j. differential geometry 61 (2002) 397-451

CHEEGER MANIFOLDS AND THE CLASSIFICATION OF BIQUOTIENTS

BURT TOTARO

Abstract

We classify the biquotient manifolds which are either rational homology spheres or Cheeger manifolds (connected sums of two rank-one symmetric spaces). Also: there are only finitely many 2-connected biquotient manifolds in each dimension.

A closed manifold is called a biquotient if it is diffeomorphic to K \ G/H for some compact Lie group G with closed subgroups K and H such that K acts freely on G/H. Every biquotient has a Riemannian metric of nonnegative sectional curvature. In fact, almost all known manifolds of nonnegative curvature are biquotients. The only known closed manifolds of nonnegative sectional curvature which were not defined in this way are those found by Cheeger [6] in 1973, and Grove and Ziller [16] in 2000.

In order to understand these constructions better, this paper analyzes which of the Cheeger and Grove-Ziller manifolds are actually diffeomorphic to biquotients. In the process, we develop a general approach to the classification of biquotient manifolds. A priori, it is hard to determine whether a given manifold is a biquotient, because there is no obvious upper bound on the dimension of the groups involved. We give a procedure which allows us to reduce the dimension of the groups needed to describe a given manifold as a biquotient. A surprising application is that Gromoll-Meyer’s example of an exotic 7-sphere which is a biquotient [15] is the only exotic sphere of any dimension which is a biquotient. More generally, we classify all biquotients which are simply connected rational homology spheres of any dimension (Theorem 6.1). The classification of biquotients which are simply connected rational homology spheres has been given independently by Kapovitch and Ziller [17]. More generally, they classify all simply connected biquotients whose rational cohomology ring is generated by one element.

Received 09/18/2002.

Another application of our general classification theory for biquotients is the precise determination of which Cheeger manifolds, the connected sums of two rank-one symmetric spaces with any orientations, are diffeomorphic to biquotients (Theorem 2.1). Some of the Cheeger manifolds were known to be biquotients, such as CP^{2}#−CP^{2} (the nontrivial S^{2}-bundle over S^{2}), but we find that many of the other Cheeger manifolds are also biquotients, such as CP^{2}#CP^{2}. On the other hand, the Cheeger manifold CP^{4}#HP^{2} is not diffeomorphic (or even homotopy equivalent) to a biquotient. The positive result that some Cheeger manifolds such as CP^{2}#CP^{2} are biquotients implies that they have nonnegatively curved Riemannian metrics with various good properties that were not clear from Cheeger’s construction. First, the new metrics are real analytic. Further, the new metrics determine a natural complexanalytic structure on the whole tangent bundle, since Aguilar [2] showed that biquotients have this property; my paper [26] has a weaker result which applies to all the Cheeger manifolds. Finally, the geodesic flow for the new metrics on the Cheeger manifolds has zero topological entropy. Paternain [23] proved the latter property for Cheeger’s metric on CP^{2}#CP^{2}, but now we know it for some metric on a large class of the Cheeger manifolds.

Finally, our general classification results imply that there are only finitely many diffeomorphism classes of 2-connected biquotients in any given dimension (Theorem 4.9). This fails for biquotients that are only simply connected already in dimension 6, as I showed in [27]. Also, the finiteness of 2-connected biquotients shows the distance between biquotients and general manifolds of nonnegative sectional curvature, since Grove and Ziller have constructed nonnegatively curved metrics on all S^{3}-bundles over S^{4}, giving infinitely many homotopy types of 2-connected 7-manifolds with nonnegative sectional curvature [16].

I would like to thank Gabriel Paternain for many useful conversations. Also, thanks to Wolfgang Ziller for several references to earlier work.

1. Notation

We begin with the equivalence of several definitions of biquotient manifolds, pointed out by Eschenburg ([9], [11]).

Lemma 1.1. The following properties of a closed smooth manifold M are equivalent:

(1) M = K\G/H for some compact Lie group G with closed subgroups K and H such that K acts freely on G/H.

(2) M = G/H for some compact Lie groups G and H together with a homomorphism H → G × G such that H acts freely on G by left and right translation, (g_{1}, g_{2})(g) := g_{1}gg_{2}^{−1}.

(3) M = G/H for some compact Lie groups G and H together with a homomorphism H → (G × G)/Z(G) such that H acts freely on G.

In (3) we are using that the center Z(G), imbedded diagonally in G × G, acts trivially on G by left and right translation.

Proof. Clearly (1) implies (2) and (2) implies (3). We prove that (3) implies (1). The point is that (G × G)/Z(G) acts transitively on G with stabilizer at 1 ∈ G equal to the diagonal subgroup G/Z(G). So, if M = G/H as in (3), then we can also describe M as in (1) by

M = (G/Z(G))\ ((G × G)/Z(G))/H.

q.e.d.

In the paper, we use definition (3) of biquotients; that is, “a biquotient G/H” will mean that G and H are compact Lie groups and we are given a homomorphism H → (G × G)/Z(G) such that H acts freely on G. When G and H are simply connected, such a homomorphism lifts uniquely to G × G.

We call a connected compact Lie group simple if it is nonabelian and every proper normal subgroup is finite. We define a simple factor of a connected compact Lie group H to be the universal cover of a simple normal subgroup of H. We write Sp(2a) for the simply connected simple group of type C_{a}, which topologists usually call Sp(a). The reason for the name Sp(2a) is that we think of each compact classical group using its standard complex representation: SU(n) is a subgroup of GL(n, C), SO(n) is a subgroup of GL(n, C), and (with the notation here) Sp(2a) is a subgroup of GL(2a, C).

By definition, the Dynkin index of a homomorphism H → G of simply connected simple groups is the integer corresponding to the homomorphism π_{3}H → π_{3}G, both groups being canonically isomorphic to Z. Dynkin computed the Dynkin index in many cases ([8], Chapter I, Section 2). Finally, we write UT(S^{n}) for the unit tangent bundle of the n-sphere, UT(S^{n}) = Spin(n + 1)/Spin(n − 1).

2. The Cheeger manifolds that are diffeomorphic to biquotients

By definition, a Cheeger manifold is the connected sum of any two rank-one symmetric spaces with any orientations. The rank-one symmetric spaces, besides the sphere which is not interesting for this purpose, are the real, complex, and quaternionic projective spaces, together with the Cayley plane associated to the octonions. The Cheeger manifolds are the only known examples of connected sums, with neither summand a homotopy sphere and at least one summand not a rational homology sphere, which admit metrics of nonnegative sectional curvature. In fact, the conjecture that manifolds of nonnegative sectional curvature are elliptic in the sense of Félix, Halperin, and Thomas [12] would imply that any connected sum which admits a metric of nonnegative sectional curvature must be roughly of the Cheeger type. Precisely, a connected sum M_{1}#M_{2} of simply connected manifolds which is elliptic, and such that neither M_{1} nor M_{2} is a k-homology sphere for a field k, must have the rings H^{∗}(M_{1}, k) and H^{∗}(M_{2}, k) both generated by a single element, as follows from Lambrechts [18], Theorem 3. Combining this with Adams and Atiyah’s results on the Hopf invariant problem [1] shows that if a connected sum of simply connected manifolds is elliptic, with neither summand a homotopy sphere and at least one summand not a rational homology sphere, then both summands have the integral cohomology ring of CP^{n}, HP^{n}, or CaP^{2}.

Theorem 2.1. The following Cheeger manifolds are diffeomor phic to biquotients. First, CP^{n}# − CP^{n}, HP^{n}# − HP^{n}, and CaP^{2}# − CaP^{2}. Next, RP^{n}#RP^{n}, RP^{2n}#CP^{n}, RP^{4n}#HP^{n}, and RP^{16}#CaP^{2}. Here RP^{n} is non-orientable for n odd and has an orientation-reversing diffeomorphism for n even, so the orientations of the summands do not matter in these cases. Next, CP^{2n}# − HP^{n}, HP^{4}# − CaP^{2}, and CP^{8}# − CaP^{2}. Finally, CP^{n}#CP^{n}, HP^{n}#HP^{n}, and CP^{4e+2}#HP^{2e+1}.

The remaining Cheeger manifolds are not even homotopy equivalent to biquotients. These are CaP^{2}#CaP^{2}, CP^{8}#CaP^{2}, HP^{4}#CaP^{2}, and CP^{4e}#HP^{2e}.

Proof. In this section, we prove only the first, positive, statement. We will prove the negative statement in Sections 7 and 8, after setting up a general classification theory of biquotients.

We begin with the cases which are straightforward generalizations of Cheeger’s observation that CP^{2}# − CP^{2} is a biquotient [6]. Let A_{k} denote the standard algebra (R, C, H, or the octonions Ca) of dimension k over R, where k = 1, 2, 4, or 8. Then, for k = 1, 2, or 4, A_{k}P^{n}# − A_{k}P^{n} is an S^{k}-bundle over A_{k}P^{n−1}, namely the biquotient manifold (S^{nk−1} × S^{k})/S^{k−1}, where S^{k−1} is a group for k = 1, 2, or 4, acting freely on S^{nk−1} and by rotations on S^{k}. For k = 8, S^{7} is not a group, but we can still describe CaP^{2}# − CaP^{2} as a biquotient,

(Spin(9) × S^{8})/Spin(8),

where Spin(8) → Spin(9) is the standard inclusion and Spin(8) acts on S^{8} ⊂ R^{9} by the direct sum of an 8-dimensional real spin representation and the trivial representation. Since Spin(9)/Spin(8) = S^{8}, this description exhibits CaP^{2}# − CaP^{2} as an S^{8}-bundle over S^{8}.

Next, we check that the connected sum of RP^{n} with any rank-one symmetric space is diffeomorphic to a biquotient. As mentioned in the theorem, orientations do not matter in this case, because RP^{n} is nonorientable for n odd and has an orientation-reversing diffeomorphism for n even. For any closed n-manifold M, the connected sum RP^{n}#M is doubly covered by M#−M. This suggests a way to view RP^{nk}#A_{k}P^{n} as a biquotient: we replace S^{k} in the above description of A_{k}P^{n}# − A_{k}P^{n} by RP^{k}. That is:

RP^{n}#RP^{n} = (S^{n−1} × S^{1})/Z/2
RP^{2n}#CP^{n} = (S^{2n−1} × RP^{2})/S^{1}
RP^{4n}#HP^{n} = (S^{4n−1} × RP^{4})/S^{3}
RP^{16}#CaP^{2} = (Spin(9) × RP^{8})/Spin(8)

The manifolds CP^{2n}, HP^{2n}, and CaP^{2} have natural orientations, corresponding to the highest power of any generator of H^{2}, H^{4}, or H^{8}, respectively. In fact, HP^{n} has a natural orientation for all n ≥ 2, but this takes more care to define. We use that the mod 3 Steenrod operation P^{1} on H^{∗}(BSU(2), F_{3}) acts by P^{1}c_{2} = c_{2}^{2}, as one checks by restricting to the torus S^{1} ⊂ SU(2). Since H^{4}(HP^{n}, Z) has a generator which is c_{2} of an SU(2)-bundle over HP^{n}, this generator c_{2} satisfies P^{1}c_{2} = c_{2}^{2} in H^{8}(HP^{n}, F_{3}). It follows that (for n ≥ 2) there is a unique generator z of H^{4}(HP^{n}, Z) such that P^{1}z = −z^{2} in H^{8}(HP^{n}, F_{3}), namely z = −c_{2}. We define the natural orientation on HP^{n} to be the one corresponding to the highest power z^{n} of this generator z.

Using this orientation, we can define one more class of Cheeger manifolds as bundles. The manifold CP^{2n}# − HP^{n} is the CP^{2}-bundle over HP^{n−1} associated to the rank-3 complex vector bundle E ⊕ C, where E is the tautological rank-2 complex vector bundle over HP^{n−1}. Thus CP^{2n}# − HP^{n} is a biquotient of the form (CP^{2} × S^{4n−1})/SU(2). We might try to imitate this construction by viewing HP^{4}# − CaP^{2} as an HP^{2}-bundle over S^{8} and CP^{8}# − CaP^{2} as a CP^{4}-bundle over S^{8}, but that turns out to be impossible. For these constructions, we would need a complex vector bundle E over S^{8} with c_{4}E equal to plus or minus the class of a point in H^{8}(S^{8}, Z), whereas in fact every complex vector bundle on S^{8} has c_{4}E a multiple of (4 − 1)! = 6, by Bott periodicity.

Instead, we construct HP^{4}# − CaP^{2} as the quotient of an S^{11}- bundle over S^{8} by a free SU(2)-action. Let S^{−} : Spin(8) → SO(8) denote one of the two spin representations. We also write S^{−} for the associated Spin(9)-equivariant real vector bundle of rank 8 over Spin(9)/Spin(8) = S^{8}. Let N be the S^{11}-bundle over S^{8} defined as the unit sphere bundle S(S^{−} ⊕ R^{4}). Define a homomorphism SU(2) → Spin(9) by SU(2) ≅ Spin(3) ⊂ Spin(9). Using this homomorphism, SU(2) acts on S^{8} and acts compatibly on the vector bundle S^{−} over S^{8}. Let SU(2) act on N by the given action on S^{8} and on the vector bundle S^{−}, and by the standard faithful representation V_{R} of SU(2) on R^{4}.

This action of SU(2) on N is free. To check this, it suffices to check that SU(2) acts freely on the S^{7}-bundle S(S^{−}) over S^{8} and on the S^{3}- bundle S(R^{4}) = S^{3} × S^{8} over S^{8}. The second statement is clear by the choice of SU(2)-action on R^{4}. To prove the first statement, first note that Spin(9) acts on S(S^{−}) ≅ S^{15} by the spin representation of Spin(9). Then use that the restriction of any spin representation of Spin(n) to Spin(n − 1) is a sum of spin representations of Spin(n − 1). Thus the action of SU(2) ≅ Spin(3) on S(S^{−}) = S^{15} is by a sum of copies of the 4-dimensional real spin representation V_{R}. It follows that SU(2) acts freely on S^{15}.

Let M be the quotient manifold N/SU(2), of dimension 16. Clearly M is a biquotient of the form (Spin(9) × S^{11})/(Spin(8) × SU(2)). By construction, M is the union of a disc bundle over S(S^{−})/SU(2) = S^{15}/SU(2) = HP^{3} and a disc bundle over S(R^{4})/SU(2) = (S^{8} × S^{3})/SU(2) = S^{8} along their common boundary, S^{15}. So M is diffeomorphic to the connected sum of HP^{4} and CaP^{2}, with some orientations. To pin down the orientations, it is convenient to compute the cohomology ring of M. Since N is an SU(2)-equivariant S^{11}-bundle over S^{8}, the quotient M is the total space of a fibration S^{11} → M → S^{8}//SU(2) up to homotopy, where S^{8}//SU(2) = (S^{8} × ESU(2))/SU(2) denotes the homotopy quotient via the given homomorphism SU(2) → Spin(9).

We compute that

H^{∗}(S^{8}//SU(2), Z) = Z[y, χ(S^{−})]/((χ(S^{−}) + y^{2})^{2} = 0),

where y in H^{4} is c_{2} of the standard representation V of SU(2) and χ(S^{−}) in H^{8} is the Euler class of the SU(2)-equivariant vector bundle S^{−} over S^{8}, with some orientation. To check this, note that the spectral sequence for the fibration S^{8} → S^{8}//SU(2) → BSU(2) degenerates, since all the cohomology is in even degrees. This implies that the cohomology of S^{8}//SU(2) has the above form, for some relation of the form χ(S^{−})^{2} + bχ(S^{−})y^{2} + cy^{4} = 0 with b, c ∈ Z. Furthermore, we have T S^{8} ⊕ R ≅ R^{9} as SU(2)-equivariant vector bundles on S^{8}. It follows that

χ(T S^{8})^{2} = c_{8}(T S^{8} ⊗_{R} C) = 0

in H^{16}(S^{8}//SU(2), Z). Also, χ(T S^{8}) has degree 2 on S^{8} while χ(S^{−}) has degree ±1 on S^{8}, so we must have χ(T S^{8}) = ±2χ(S^{−}) + dy^{2} in H^{8}(S^{8}//SU(2), Z) for some integer d. So (±χ(S^{−}) + (d/2)y^{2})^{2} = 0 in H^{16}(S^{8}//SU(2), Q). By what we know about the form of the relation, d/2 must be an integer a, and we have

(±χ(S^{−}) + ay^{2})^{2} = 0

in H^{16}(S^{8}//SU(2), Z). Finally, the action of SU(2) = Spin(3) ⊂ Spin(9) on S^{8} preserves a 2-sphere, and so we have an inclusion BS^{1} ≃S^{2}//SU(2) → S^{8}//SU(2). Let x be the standard generator of the polynomial ring H^{∗}(BS^{1}, Z) in degree 2. We compute that the restriction map takes y = c_{2}V to −x^{2} and χ(S^{−}) to ±x^{4}, using that the restriction of the spin representation S^{−} to S^{1} = Spin(2) ⊂ Spin(8) is the direct sum of 4 copies of the standard 2-dimensional real representation of S^{1}. Therefore, for a suitable orientation on S^{−}, the relation in H^{16}(S^{8}//SU(2), Z) must be

(χ(S^{−}) + y^{2})^{2} = 0.

Since M is an S^{11}-bundle over S^{8}//SU(2), we have one more relation in H^{12}(M, Z), saying that the Euler class of this S^{11}-bundle is zero. The S^{11}-bundle is S(S^{−}⊕V_{R}), and so its Euler class is χ(S^{−}⊕V_{R}) = χ(S^{−})y in H^{12}(S^{8}//S^{1}, Z). Thus M has cohomology ring

H^{∗}(M, Z) = Z[y, χ(S^{−})]/((χ(S^{−}) + y^{2})^{2} = 0, χ(S^{−})y = 0)
= Z[y, χ(S^{−})]/(χ(S^{−})^{2} + y^{4} = 0, χ(S^{−})y = 0).

This is the cohomology ring of HP^{4}# − CaP^{2}, not of HP^{4}#CaP^{2}. So the biquotient M is diffeomorphic to HP^{4}# − CaP^{2}.

The proof that CP^{8}# − CaP^{2} is diffeomorphic to a biquotient M is completely analogous: it is the quotient of the S^{9}-bundle S(S^{−} ⊕ R^{2}) over S^{8} by a free S^{1}-action. Here S^{1} acts on S^{8} and on the vector bundle S^{−} by the homomorphism S^{1} ≅ Spin(2) ⊂ Spin(9), and on R^{2} by the standard real representation of S^{1}. Thus M is a biquotient of the form (Spin(9) × S^{9})/(Spin(8) × S^{1}).

Finally, we come to the most surprising examples of biquotients, starting with CP^{n}#CP^{n}. The manifold CP^{n} has an orientation-revers-ing diffeomorphism from n odd, and so CP^{n}#CP^{n} for n odd is diffeomorphic to CP^{n}# − CP^{n}. For n even, we will show that CP^{n}#CP^{n} is diffeomorphic to a biquotient (S^{3} × S^{2n−1})/(S^{1})^{2}, for a free isometric action of (S^{1})^{2} on S^{3} × S^{2n−1}. The action is defined by the following homomorphism from (S^{1})^{2} to the maximal torus (S^{1})^{2} × (S^{1})^{n} of SO(4) × SO(2n):

(x, y) ↦ ((x, y), (xy^{−1}, xy, …, xy)).

It is straightforward to check that this action of (S^{1})^{2} on S^{3} × S^{2n−1} is free. Let M be the quotient manifold. We use that the action of (S^{1})^{2} on S^{3} has cohomogeneity one, with trivial generic stabilizer group and stabilizers at the two special orbits equal to the two factors S^{1}. It follows that M is the union of a 2-disc bundle over S^{2n−1}/S^{1} = CP^{n−1} (corresponding to the action of the first factor S^{1}) and a 2-disc bundle over S^{2n−1}/S^{1} = CP^{n−1} (corresponding to the second S^{1}) along their common boundary, which is a sphere S^{2n−1}. So M is the connected sum of two copies of CP^{n}, with some orientations. To work out the orientations, it is convenient to compute the cohomology ring of M.

The quotient manifold M fits into a fibration

S^{3} × S^{2n−1} → M → (BS^{1})^{2}.

Here (BS^{1})^{2} has cohomology ring Z[u, v] with u and v in H^{2}. From the description of the (S^{1})^{2}-action, we read off that the Euler classes of the S^{3}-bundle and S^{2n−1}-bundle over (BS^{1})^{2} are uv and (u − v)(u + v)^{n−1}. Since these form a regular sequence in the polynomial ring H^{∗}((BS^{1})^{2}, Z), we have

H^{∗}(M, Z) = Z[u, v]/(uv = 0, (u − v)(u + v)^{n−1} = 0)
= Z[u, v]/(uv = 0, u^{n} = v^{n}).

This is the cohomology ring of CP^{n}#CP^{n}, and (for n even) not that of CP^{n}# − CP^{n}. Therefore the biquotient M is diffeomorphic to CP^{n}#CP^{n}.

We next show that HP^{n}#HP^{n} is diffeomorphic to a biquotient. Imitating the construction for CP^{n}#CP^{n} might suggest identifying HP^{n}#HP^{n} with a biquotient (S^{7} × S^{8n−1})/SU(2)^{2}. That works for n odd, but not for n even. For n even, we have to consider a more general type of biquotient, (Sp(4) × S^{4n−1})/SU(2)^{3}. Here the group SU(2)^{3} acts on Sp(4) by a homomorphism SU(2)^{3} → Sp(4)^{2}. To define this, first let V_{i} for i = 1, 2, 3 denote the standard 2-dimensional complex representation of the ith factor of SU(2). Then the homomorphism SU(2)^{3} → Sp(4)^{2} is defined by (V_{1} ⊕ V_{2}, V_{3} ⊕ C^{2}). Also, let W_{12} : SU(2)^{2} → SO(4) be the natural double covering, viewed as a 4-dimensional real representation of the first two copies of SU(2), and let (V_{3})_{R} : SU(2) → SO(4) be the natural real faithful representation of the third copy of SU(2). We define the action of SU(2)^{3} on S^{4n−1} by the homomorphism SU(2)^{3} → SO(8n) defined by (V_{3})_{R}^{⊕n−1} ⊕ W_{12}.

The action of SU(2)^{3} on Sp(4) has cohomogeneity one, with trivial generic stabilizer group and with stabilizers at the two special orbits both isomorphic to SU(2), the first conjugate to the subgroup {(x, 1, x)} in SU(2)^{3} and the second conjugate to {(1, x, x)}. These two subgroups act freely on S^{4n−1}, and so SU(2)^{3} acts freely on Sp(4) × S^{4n−1}. Let M be the quotient manifold. Using the description of the SU(2)^{3}- action on Sp(4), we see that M is the union of two disc bundles over S^{4n−1}/SU(2) = HP^{n−1} along their common boundary, S^{4n−1}. Therefore M is the connected sum of two copies of HP^{n}, with some orientations.

To determine the relevant orientations, it suffices for n even to compute the cohomology ring of M. For n odd, the manifolds HP^{n}#HP^{n} and HP^{n}# − HP^{n} have isomorphic cohomology rings. In that case we will also need a mod 3 Steenrod operation to see that M is diffeomorphic to HP^{n}#HP^{n} rather than HP^{n}# − HP^{n}. First, we compute the cohomology ring of the homotopy quotient Sp(4)//SU(2)^{3}, with respect to the above action of SU(2)^{3} on Sp(4). Write z_{i} = −c_{2}V_{i}, for i = 1, 2, 3, which are generators in H^{4} of the polynomial ring H^{∗}(BSU(2)^{3}, Z). The generators c_{2} and c_{4} of H^{∗}(BSp(4), Z) pull back under the left homomorphism SU(2)^{3} → Sp(4) to −z_{1} − z_{2} and z_{1}z_{2}, and under the right homomorphism SU(2)^{3} → Sp(4) to −z_{3} and 0. Therefore, using the Eilenberg-Moore spectral sequence as suggested by Singhof [25], the cohomology of Sp(4)//SU(2)^{3} is

Z[z_{1}, z_{2}, z_{3}]/(z_{1} + z_{2} = z_{3}, z_{1}z_{2} = 0).

The manifold M is an S^{4n−1}-bundle over Sp(4)//SU(2)^{3}, up to homotopy. The Euler class of this bundle, (V_{3})_{R}^{⊕n−1}⊕W_{12}, is ±(−z_{3})^{n−1}(−z_{1}+ z_{2}), and so M has cohomology ring

Z[z_{1}, z_{2}, z_{3}]/(z_{1} + z_{2} = z_{3}, z_{1}z_{2} = 0, z_{3}^{n−1}(z_{1} − z_{2}) = 0)
= Z[z_{1}, z_{2}]/(z_{1}z_{2} = 0, z_{1}^{n} = z_{2}^{n}).

This is the cohomology ring of HP^{n}#HP^{n}, and for n even it is not the cohomology ring of HP^{n}#−HP^{n}. So M is diffeomorphic to HP^{n}#HP^{n} for n even. For n odd, n ≥ 3, we also need to observe that the classes z_{i} are distinguished from their negatives by the fact that P^{1}z_{i} = −z_{i}^{2} in H^{8}(M, F_{3}). By definition of the natural orientation on HP^{n} for n ≥ 2, this means that M is diffeomorphic to HP^{n}#HP^{n} for all n ≥ 2, not to HP^{n}#−HP^{n}. The case n = 1 is trivial, since HP^{n}#HP^{n} = S^{4}#S^{4} = S^{4}.

The last Cheeger manifolds which are diffeomorphic to biquotients are the manifolds CP^{4e+2}#HP^{2e+1}. The relevant biquotient M has the form M = (S^{5} × S^{8e+3})/(S^{1} × SU(2)). To be more explicit, let L be the standard 1-dimensional complex representation of S^{1}, and let V be the standard 2-dimensional complex representation of SU(2). Then we let S^{1} × SU(2) act on S^{5} as the unit sphere in V ⊕ L, and on S^{4n−1} as the unit sphere in (V ⊗_{C} L)^{⊕2e+1}. The action of S^{1} × SU(2) on S^{5} has cohomogeneity one, with trivial generic stabilizer and with stabilizers at the two special orbits conjugate to the two factors S^{1} and SU(2), respectively. Both of these subgroups act freely on S^{8e+3}, and so S^{1} × SU(2) acts freely on S^{5} × S^{8e+3}. Let M be the quotient manifold.

From the action of S^{1} × SU(2) on S^{5}, we see that M is the union of a disc bundle over S^{8e+3}/S^{1} = CP^{4e+1} and a disc bundle over S^{8e+3}/SU(2) = HP^{2e} along their common boundary, S^{8e+3}. It follows that M is the connected sum of CP^{4e+2} and HP^{2e+1} with some orientations.

To show that M is diffeomorphic to CP^{4e+2}#HP^{2e+1} rather than to CP^{4e+2}# − HP^{2e+1}, we compute the cohomology ring and a mod 3 Steenrod operation on M. Since M = S(V ⊕L)×S((V ⊗L)^{⊕2e+1})/(S^{1}× SU(2)), we can view M as an (S^{5} × S^{8e+3})-bundle over BS^{1} × BSU(2). Here BS^{1} × BSU(2) has cohomology ring Z[x, z], where we let x = c_{1}L and z = −c_{2}V. The vector bundles V ⊕ L and (V ⊗ L)^{⊕2e+1} on BS^{1} × BSU(2) have Euler classes (−z)x and (x^{2} − z)^{2e+1}. So M has cohomology ring

Z[x, z]/(−xz = 0, (x^{2} − z)^{2e+1} = 0)
= Z[x, z]/(xz = 0, x^{4e+2} = z^{2e+1}).

Here the element z in H^{4}(M, Z) is distinguished from its negative, for e ≥ 1, by the property that P^{1}z = −z^{2} in H^{8}(M, F_{3}). By definition of the natural orientation on HP^{2e+1} for e ≥ 1, M is diffeomorphic to CP^{4e+2}#HP^{2e+1}, not to CP^{4e+2}# − HP^{2e+1}.

By contrast, the analogous calculation of the cohomology ring shows that the biquotient

(S(V ⊕ L) × S((V ⊗ L)^{⊕2e}))/(S^{1} × SU(2))

is diffeomorphic to CP^{4e}# − HP^{2e}, not to CP^{4e}#HP^{2e}. This will be used in Section 8.

3. Simplifying the description of a given biquotient manifold

Here is an elementary but essential beginning to our simplification of the description of a given biquotient manifold. (Throughout the paper, G and H will denote compact Lie groups.)

Lemma 3.1. Let M be a simply connected biquotient manifold. Then we can write M = G/H for some simply connected group G and connected group H acting on G by a homomorphism H → (G × G)/Z(G). If M is 2-connected, then H is simply connected and H acts on G by a homomorphism H → G × G.

Proof. Since M is connected, we can write M as a biquotient G/H with G connected. Since M is simply connected, the long exact sequence of the fibration H → G → M,

π_{1}H → π_{1}G → π_{1}M → π_{0}H → π_{0}G,

shows that H is connected and π_{1}H → π_{1}G is surjective. Let C be the kernel of π_{1}H → π_{1}G, a finitely generated abelian group.

Let G̃ and H̃ be the universal covers of G and H. We can identify π_{1}H with the kernel of H̃ → H, and so C is a central subgroup of H̃. We have G̃ ≅ K_{G} × R^{a} and H̃ ≅ K_{H} × R^{b} for some simply connected compact Lie groups K_{G} and K_{H}. The given homomorphism H → (G × G)/Z(G) lifts to a homomorphism H̃ → G̃ × G̃. The resulting action of H̃ on G̃ is trivial on the subgroup C, and H̃/C acts freely on G̃ with M = G̃/(H̃/C).

Here G̃ = K_{G} × R^{a}, and the action of H̃/C on G̃ is the product of an action on K_{G} and an action on R^{a}. Furthermore, since the group R^{a} is abelian, H̃/C acts on R^{a} by translations (left or right translations being the same). Since H̃/C is a connected group and the quotient is compact, H̃/C must act transitively on R^{a}. Let L be the kernel of the action of H̃/C on R^{a}. Then L must act freely on K_{G}, with M = K_{G}/L. Here K_{G} is a simply connected compact Lie group. Also, L must be connected by the long exact sequence

π_{1}M → π_{0}L → π_{0}K_{G}.

Finally, if M is 2-connected, then the same long exact sequence shows that L is simply connected. In this case, the homomorphism L → (K_{G}×K_{G})/Z(G) lifts uniquely to a homomorphism L → K_{G}×K_{G}. q.e.d.

Since we intend to study simply connected biquotient manifolds, Lemma 3.1 tells us that we can assume G is simply connected, and thus a product of simply connected simple groups. Thus, much of the complexity of more general compact Lie groups is avoided.

The main method of simplifying the description of a given biquotient is the following easy observation.

Lemma 3.2. Let H be a compact Lie group acting on manifolds X_{1} and X_{2} such that H acts transitively on X_{1} and H acts freely on X_{1} × X_{2}. Let K ⊂ H be the stabilizer of some point in X_{1}. Then the quotient manifold (X_{1}×X_{2})/H is diffeomorphic to X_{2}/K, where K acts freely on X_{2} by the restriction of the action of H.

Proof. This is clear by identifying X_{1} with H/K. q.e.d.

Applying this method of simplification to biquotients gives the following fundamental result.

Lemma 3.3. Let M be a simply connected biquotient manifold. Then we can write M = G/H such that G is simply connected, H is connected, and H does not act transitively on any simple factor of G.

Proof. By Lemma 3.1, we can write M as a biquotient G/H with G simply connected and H connected. If H acts transitively on some simple factor of G, then Lemma 3.2 allows us to remove that factor of G while replacing H by a subgroup. The exact sequence π_{1}M → π_{0}H → π_{0}G shows that the new subgroup H is still connected. By induction on the number of simple factors of G, we can arrange that H does not act transitively on any simple factor of G. q.e.d.

Convention 3.4. From now on, we will only consider simply connected biquotient manifolds, and in writing M = G/H we will assume that G and H satisfy the properties listed in Lemma 3.3.

4. Bounding G in terms of M

As mentioned in Convention 3.4, for the rest of the paper we only consider simply connected biquotient manifolds. We can and do assume that every such manifold M is written M = G/H with G simply connected, H connected, and H not acting transitively on any simple factor of G. In this section, we will show how these properties determine G up to finitely many possibilities in terms of the rational homotopy groups of M.

The following lemma will not be needed in this section, but is included here for later use.

Lemma 4.1. Let M = G/H be a simply connected biquotient manifold. If in addition the rational homotopy group π_{3}M_{Q} is zero, then each simple factor H_{1} of H acts trivially on all factors of G isomorphic to H_{1}.

Proof. By the long exact sequence of the fibration H → G → M, since π_{3}M_{Q} = 0, the homomorphism π_{3}H_{Q} → π_{3}G_{Q} is surjective. In particular, for each simple factor G_{1} of G, π_{3}H_{Q} → π_{3}(G_{1})_{Q} is surjective. Suppose that some simple factor H_{1} of H which is isomorphic to G_{1} acts nontrivially on G_{1}; we will derive a contradiction. Here, a priori, H_{1} can act by left and right translation on G_{1}. If H_{1} acts nontrivially on only one side of G_{1}, thus by a nontrivial homomorphism H_{1} → G_{1}, then this homomorphism must be an isomorphism, since H_{1} is simple. In particular, H_{1} acts transitively on G_{1}, which contradicts Convention 3.4. Therefore, H_{1} must act nontrivially on both sides of G_{1}. Then the two homomorphisms from H_{1} to G_{1} must both be isomorphisms, using simplicity of H_{1} again. We now use that any automorphism of a simply connected simple group G_{1} acts trivially on π_{3}(G_{1}) ≅ Z, as follows from the proof of this isomorphism using the Killing form. The homomorphism π_{3}(H_{1})_{Q} → π_{3}(G_{1})_{Q} is the difference of the homomorphisms given by the two homomorphisms H_{1} → G_{1}, and so it is zero. Furthermore, since the two homomorphisms H_{1} → G_{1} both have only finite centralizer, there is no room for the rest of H to act on G_{1}; in other words, H acts on G_{1} through a quotient group isogenous to H_{1}. It follows that the homomorphism π_{3}H_{Q} → π_{3}(G_{1})_{Q} is zero, contradicting what we know from π_{3}M_{Q} = 0. Thus, each simple factor H_{1} of H must act trivially on simple factors of G isomorphic to H_{1}. q.e.d.

To find more precise information on G, we need the classification of the simple compact Lie groups. In particular, we use that a simple group G has rational homotopy groups concentrated in odd degrees 2d − 1, and we call the numbers d that occur the degrees of G. Particularly important for us is the maximal degree of G, which we call d(G). It is also called the Coxeter number of G. The degrees of G are well-known in many contexts: we can also say that H^{∗}(G, Q) is an exterior algebra with generators in degrees 2d − 1 where d runs over the degrees of G, or that the degrees of G are the degrees of the generators of the ring of invariants of the Weyl group acting on its reflection representation. We tabulate the degrees of the simple groups here, following Bourbaki [5] or Gorbatsevich-Onishchik ([14], Table 1, p. 127). To avoid repetitions, one can assume that A_{l} has l ≥ 1, B_{l} has l ≥ 3, C_{l} has l ≥ 2, and D_{l} has l ≥ 4.

Table 4.2

A_{l} : 2, 3, …, l + 1
B_{l} : 2, 4, 6, …, 2l
C_{l} : 2, 4, 6, …, 2l
D_{l} : 2, 4, 6, …, 2l − 2; l
G_{2} : 2, 6
F_{4} : 2, 6, 8, 12
E_{6} : 2, 5, 6, 8, 9, 12
E_{7} : 2, 6, 8, 10, 12, 14, 18
E_{8} : 2, 8, 12, 14, 18, 20, 24, 30.

Using Table 4.2 and the known low-dimensional representations of each group, Onishchik proved the following result [20]. He later gave a more systematic proof, using reflection groups [21].

Lemma 4.3. Let H → G be a nontrivial homomorphism of simply connected simple groups. Then the maximal degrees satisfy d(H) ≤ d(G). Moreover, if d(H) = d(G), then either H → G is an isomorphism or G/H is one of the following homogeneous spaces. On the right we show the degrees of G and H.

Spin(2n)/Spin(2n − 1)

= S^{2n−1}, n ≥ 4  2, 4, 6, …, 2n − 2; n 2, 4, 6, …, 2n − 2
SU(2n)/Sp(2n), n ≥ 2  2, 3, 4, …, 2n 2, 4, 6, … 2n
Spin(7)/G_{2} = S^{7}  2, 4, 6 2, 6
Spin(8)/G_{2} = S^{7} × S^{7}  2, 4, 4, 6 2, 6
E_{6}/F_{4}  2, 5, 6, 8, 9, 12 2, 6, 8, 12.

In all these cases except Spin(8)/Spin(7) = S^{7}, there is a unique conjugacy class of nontrivial homomorphisms H → G; in the case Spin(8)/Spin(7), there are three conjugacy classes which are equivalent under outer automorphisms of Spin(8). Also, in all the above cases, the centralizer of H in G is finite.

The following result shows how to apply Lemma 4.3 to biquotients M = G/H, although it is only a step on the way to the more precise Theorem 4.8. For a simple factor G_{1} of G, we say that a degree d of G_{1} is killed by H if the homomorphism π_{2d−1}H_{Q} → π_{2d−1}(G_{1})_{Q} associated to the action of H on G_{1} is nonzero.

Lemma 4.4. Let M = G/H be a simply connected biquotient, written according to Convention 3.4. Let G_{1} be a simple factor of G such that the maximal degree of G_{1} is killed by H. Then either there is a simple factor H_{1} of H such that H_{1} acts nontrivially on exactly one side of G_{1} by one of the homomorphisms in Lemma 4.3, so that G_{1}/H_{1} is one of Spin(2n)/Spin(2n − 1) = S^{2n−1}, SU(2n)/Sp(2n), Spin(7)/G_{2} = S^{7}, Spin(8)/G_{2} = S^{7} × S^{7}, or E_{6}/F_{4}; or G_{1} is isomorphic to SU(2n + 1) for some n and there is a simple factor H_{1} of H also isomorphic to SU(2n + 1) which acts on G_{1} by h(g) = hgh^{t}. The SU(2n + 1) case cannot occur if π_{3}M_{Q} = 0.

Proof. We are given a simple factor G_{1} of G such that the maximal degree of G_{1} is killed by H. It follows that there must be a simple factor H_{1} of H which kills the maximal degree d of G_{1}. Since H_{1} is simply connected, the action of H_{1} on G_{1} is given by a homomorphism H_{1} → G_{1} × G_{1}, and the resulting linear map π_{2d−1}(H_{1})_{Q} → π_{2d−1}(G_{1})_{Q} ≅ Q is nonzero. This linear map is the difference of the two linear maps associated to the two homomorphisms H_{1} → G_{1} (on the left and right), so at least one of those two linear maps is nonzero. By Lemma 4.3, either H_{1} is isomorphic to G_{1} or (G_{1}, H_{1}) is one of the pairs (Spin(2n), Spin(2n−1)), (SU(2n), Sp(2n)), (Spin(7), G_{2}), (Spin(8), G_{2}), or (E_{6}, F_{4}).

Suppose first that (G_{1}, H_{1}) is one of these 5 pairs. If H_{1} acts nontrivially on both sides of G_{1}, then in all cases except (Spin(8), Spin(7)), Lemma 4.3 implies that the two homomorphisms H_{1} → G_{1} are conjugate, so the resulting map π_{∗}(H_{1})_{Q} → π_{∗}(G_{1})_{Q} (the difference of the left and right maps) is 0, contradicting the fact that H_{1} kills the top degree of G_{1}. Even in the case (Spin(8), Spin(7)), we compute that the outer automorphism group of Spin(8) acts trivially on the top degree, 6, of Spin(8), that is, on π_{11}Spin(8)_{Q}. It follows that if Spin(7) acts nontrivially on both sides of Spin(8), then the left and right homomorphisms Spin(7) → Spin(8) give the same linear map into the top degree of Spin(8), and so the action of Spin(7) of Spin(8) cannot kill the top degree of Spin(8), a contradiction. Thus, in all these cases, H_{1} acts nontrivially on only one side of G_{1}, by one of the homomorphisms in Lemma 4.3. The lemma is proved in this case.

The remaining case is where H_{1} is isomorphic to G_{1}. Clearly this cannot occur if π_{3}M_{Q} = 0, by Lemma 4.1. In general, if H_{1} acts nontrivially on only one side of G_{1}, then it must act by an isomorphism H_{1} → G_{1}. So H acts transitively on G_{1}, contrary to Convention 3.4. Therefore H_{1} must act nontrivially on both sides of G_{1}, clearly by two isomorphisms H_{1} → G_{1}. We are given that H_{1} kills the top degree of G_{1}, so these two isomorphisms H_{1} → G_{1} must give different linear maps into the top degree of π_{∗}(G_{1})_{Q}. So the outer automorphism group of G_{1} must act nontrivially on the top degree of G_{1}. Of the simply connected simple groups, only A_{n}, D_{n}, and E_{6} have nontrivial outer automorphism group, and only in the case G_{1} = SU(2n + 1) = A_{2n} does the outer automorphism group act nontrivially on the top degree of G_{1}. (The outer automorphism group Z/2 of SU(n), n ≥ 3, acts by the identity on the even degrees of SU(n) and by −1 on the odd degrees. A topological way to see this is to identify π_{∗}G_{Q} with the dual to the vector space H^{>0}(BG, Q)/(H^{>0}·H^{>0}), and use in the case G = SU(n) that the outer automorphism E ↦ E^{∗} acts on Chern classes by c_{i} ↦ (−1)^{i}c_{i}.) So we must have G_{1} = SU(2n + 1), with H_{1} = SU(2n + 1) acting on G_{1} by the identity on one side and by the outer automorphism x ↦ (x^{t})^{−1} on the other. q.e.d.

In order to have strong restrictions on the simple factors of G in terms of the rational homotopy groups of M = G/H, we need to analyze more completely the way H acts on simple factors of G isomorphic to Spin(2n), SU(2a), Spin(7), Spin(8), or E_{6}. In analogy with Lemmas 4.3 and 4.4, we will first formulate a general statement on the subgroups of these groups, and then apply it to biquotients. Before that, we need some simple topological results.

Lemma 4.5. Let H and G be connected compact Lie groups, and let f : H → G be any continuous map. If f_{∗} : π_{∗}H_{Q} → π_{∗}G_{Q} is surjective, then f is surjective.

Proof. Both H and G have the rational homotopy type of products of odd-dimensional spheres. So H^{∗}(H, Q) and H^{∗}(G, Q) are both exterior algebras, and the assumption means that the homomorphism

f^{∗} : H^{>0}(G, Q)/(H^{>0} · H^{>0}) → H^{>0}(H, Q)/(H^{>0} · H^{>0})

is injective. Thus H^{∗}(H, Q) is the exterior algebra generated by the generators of H^{∗}(G, Q) together with some other generators. So H^{∗}(G, Q) → H^{∗}(H, Q) is injective. Since G is a closed orientable manifold, it follows that H → G is surjective. q.e.d.

Corollary 4.6. Let H and G be connected compact Lie groups, with an action of H on G by a homomorphism H → (G × G)/Z(G). If the associated homomorphism π_{∗}H_{Q} → π_{∗}G_{Q} is surjective, then H acts transitively on G.

Proof. We are given that the homomorphism associated to the orbit map f : H → G of some point in G is surjective on rational homotopy groups. By Lemma 4.5, H → G is surjective. That is, H acts transitively on G. q.e.d.

We now apply Corollary 4.6 to get information on the subgroups of the groups occurring in Lemma 4.3. (We also use the classification of simple Lie groups in the following proof, but it is more pleasant to use Corollary 4.6 when possible.)

Lemma 4.7. Let φ : H → Spin(2n) be a homomorphism from a simply connected simple group such that the homomorphism

π_{2n−1}H_{Q} → π_{2n−1}Spin(2n)_{Q}/π_{2n−1}Spin(2n − 1)_{Q} ≅ Q

is not zero. Then H acts transitively on the sphere Spin(2n)/Spin(2n−1) = S^{2n−1}.

Next, let G/K be one of the other homogeneous spaces from Lemma 4.3: SU(2n)/Sp(2n) with n ≥ 2, Spin(7)/G_{2} = S^{7}, Spin(8)/G_{2} = S^{7} × S^{7}, or E_{6}/F_{4}. Let H → G be a homomorphism from a simply connected compact Lie group H which kills the second-largest degree of G. (In the case G/K = Spin(8)/G_{2}, where Spin(8) has degrees 2, 4, 4, 6, we assume that both degrees 4 of Spin(8) are killed by H.) Then H acts transitively on G/K.

Proof. In the cases where G/K is one of Spin(2n)/Spin(2n − 1) = S^{2n−1}, Spin(7)/G_{2} = S^{7}, or Spin(8)/G_{2} = S^{7} × S^{7}, the assumption implies that the action of H ×K on G is surjective on rational homotopy groups. By Corollary 4.6, H × K acts transitively on G. Equivalently, H acts transitively on G/K.

Next, let G/K = SU(2n)/Sp(2n), n ≥ 2, and suppose that H kills the second-largest degree, 2n − 1, of SU(2n). We can replace H by one of its simple factors without changing this property. If H is isomorphic to G, then the homomorphism H → G is an isomorphism, and so H acts transitively on G/K. Otherwise, by Lemma 4.3, H has maximal degree at most that of SU(2n), which is 2n, and if equality holds then H = Sp(2n). But Sp(2n) does not kill the degree 2n − 1 of SU(2n). So H must have maximal degree 2n − 1. By Table 4.2, the only simple group with maximal degree an odd number is H = SU(2n − 1). Since n ≥ 2, any nontrivial homomorphism SU(2n−1) → SU(2n) is equivalent to the standard inclusion by some automorphism of SU(2n − 1). Then SU(2n − 1) acts transitively on SU(2n)/Sp(2n), because Sp(2n) acts transitively on SU(2n)/SU(2n − 1) = S^{4n−1}.

Finally, suppose that G/K = E_{6}/F_{4} and H kills the second-largest degree, 9, of E_{6}. We can replace H by one of its simple factors without changing this property. Then H has rank at most the rank 6 of E_{6} and has 9 as a degree. By Table 4.2, it follows that H is isomorphic to E_{6}. So the given homomorphism H → E_{6} must be an isomorphism, and in particular H acts transitively on E_{6}/F_{4}. q.e.d.

We now apply this result on subgroups to deduce strong information on the classification of biquotient manifolds. For a given biquotient M = G/H, we say that a given simple factor G_{i} of G contributes degree d_{i} to M if the homomorphism π_{2d_{i}−1}H_{Q} → π_{2d_{i}−1}(G_{i})_{Q} is not surjective. If G = ∏ G_{i} and every factor G_{i} contributes some degree d_{i} to M, then π_{2d−1}M_{Q} = π_{2d−1}G_{Q}/π_{2d−1}H_{Q} has dimension at least equal to the number of simple factors G_{i} with d_{i} = d.

Theorem 4.8. Let M = G/H be a simply connected biquotient manifold, written using Convention 3.4. Let G_{1} be any simple factor of G. Then at least one of the following holds:

(1) G_{1} contributes its maximal degree to M.

(2) G_{1} contributes its second-largest degree to M, and there is a simple factor H_{1} of G_{1} which acts nontrivially on exactly one side of G_{1}, with G_{1}/H_{1} isomorphic to one of the homogeneous spaces SU(2n)/Sp(2n) with n ≥ 2, Spin(7)/G_{2} = S^{7}, Spin(8)/G_{2} = S^{7} × S^{7}, or E_{6}/F_{4}. The second-largest degree of G_{1} is, respectively, 2a − 1, 4, 4, or 9. (In the case G_{1} = Spin(8), which has degrees 2, 4, 4, 6, the claim is only that G_{1} contributes at least one degree 4 to M.)

(3) G_{1} ≅ Spin(2n) with n ≥ 4 contributes its degree n to M, and there is a simple factor H_{1} ≅ Spin(2n − 1) which acts nontrivially on exactly one side of G_{1}, by the standard inclusion, with Spin(2n)/Spin(2n − 1) = S^{2n−1}.

(4) G_{1} ≅ SU(2n+1) contributes degrees 2, 4, 6, …, 2n to M, and there is a simple factor H_{1} ≅ SU(2n + 1) of G_{1} which acts on G_{1} by h(g) = hgh^{t}.

Proof. Let G_{1} be a simple factor of G. Suppose that (1) does not hold, in other words that the maximal degree of G_{1} is killed by H. By Lemma 4.4, there is a simple factor H_{1} of H such that either H_{1} acts nontrivially on exactly one side of G_{1} by one of the homomorphisms listed in (2) or (3) above, or G_{1} = H_{1} = SU(2n + 1) and H_{1} acts on G_{1} as in (4). The remaining point is to show that G_{1} contributes the degrees to M that we have claimed.

In cases (2) and (3), H_{1} has finite centralizer in G_{1} by Lemma 4.3, so the rest of H can act on G_{1} only on the other side from H_{1}. Since H_{1} and the rest of H together do not act transitively on G_{1}, by Convention 3.4, Lemma 4.7 shows that G_{1} contributes its degree n to M if G_{1}/H_{1} = Spin(2n)/Spin(2n − 1) = S^{2n−1}, or its second-largest degree to M if G_{1}/H_{1} is one of SU(2n)/Sp(2n), Spin(7)/G_{2} = S^{7}, Spin(8)/G_{2} = S^{7} × S^{7}, or E_{6}/F_{4}.

In case (4), since H_{1} = SU(2n + 1) acts with finite centralizer on both sides of G_{1} = SU(2n + 1), no other factor of H can act on G_{1}. So the image of π_{∗}H_{Q} → π_{∗}(G_{1})_{Q} is equal to the image of π_{∗}(H_{1})_{Q} → π_{∗}(G_{1})_{Q}. This homomorphism is the difference of the identity map on π_{∗}(G_{1})_{Q} with the map given by the outer automorphism g ↦ (g^{t})^{−1}, which acts by 1 on the even degrees and by −1 on the odd degrees (as shown in the proof of Lemma 4.4). So the image of π_{2d−1}(H_{1})_{Q} → π_{2d−1}(G_{1})_{Q} is zero for d even. That is, G_{1} contributes all its even degrees 2, 4, …, 2n to M. q.e.d.

Theorem 4.8 implies the following important qualitative statement on the classification of biquotients. The analogous result for homogeneous spaces is easy and probably well-known. It is perhaps surprising that the following statement requires the detailed classification work we have done in this section, but that seems to be true, at least for now.

Theorem 4.9. There are only finitely many diffeomorphism classes of 2-connected biquotient manifolds of a given dimension.

Theorem 4.9 is vaguely reminiscent of the Petrunin-Tuschmann theorem, which says in particular that for any number C, there are only finitely many diffeomorphism classes of 2-connected closed Riemannian manifolds with curvature 0 ≤ K ≤ C and diameter 1 [24]. But there is probably no way to actually deduce Theorem 4.9 from the Petrunin-Tuschmann theorem, since there is no obvious upper bound on the curvature of 2-connected biquotients until one goes through the proof of Theorem 4.9. In fact, the discussion after Theorem 1.1 in my paper [27] shows that there can be no upper bound on the curvature of simply connected biquotients of dimension 6, if one fixes their diameter to be 1.

Proof of Theorem 4.9. Any biquotient manifold M is rationally elliptic; that is, all but finitely many of the rational homotopy groups of M are zero. So, writing n for the dimension of M, the odd-degree rational homotopy groups π_{2d−1}M_{Q} are zero for d > n, and the total dimension of the odd-degree rational homotopy groups of M is at most n, by Friedlander and Halperin ([13], p. 434). Since M is simply connected, we can write M as a biquotient G/H according to Convention 3.4. In particular, G is simply connected and, since M is 2-connected, H is also simply connected. The essential point is Theorem 4.8, which implies that each simple factor of G contributes at least one degree to M, and that one such degree is at least half the maximal degree of G. Therefore the number of simple factors of G is at most the total dimension of π_{odd}M_{Q} and hence at most n, and the maximal degree of each simple factor of G is at most 2n. Thus G, being a product of simply connected simple groups, is determined up to finitely many possibilities by n. Also, H is a simply connected group of dimension at most that of G, so H and the homomorphism H → (G × G)/Z(G) (up to conjugacy) are determined up to finitely many possibilities. q.e.d.

5. Further general results on the classification of biquotients

In this section we continue the previous section’s method: we classify subgroups of compact Lie groups with certain properties, and apply the results to the classification of biquotients. The main result of this section is Theorem 5.3, which describes the possible simple factors of G in a biquotient M = G/H which contribute only their top degree to M. (This terminology is defined before Theorem 4.8.)

We begin by stating a classification of certain subgroups of Spin(2n).

Lemma 5.1. Let φ : H → Spin(2n), n ≥ 4, be a homomorphism from a simply connected simple group H such that the linear map

π_{2n−1}H_{Q} → π_{2n−1}Spin(2n)_{Q}/π_{2n−1}Spin(2n − 1)_{Q} ≅ Q

is not zero. Then φ is either an isomorphism or one of the following homomorphisms, up to the standard Z/2 group of outer automorphisms of Spin(2n): SU(n) ↪ Spin(2n), Sp(2a) ↪ Spin(4a), the spin representation Spin(7) ↪ Spin(8), or the spin representation Spin(9) ↪ Spin(16).

This is straightforward to prove using the known degrees and lowdimensional representations of all the simple groups. Alternatively, we can deduce it from Borel’s classification of groups which act linearly and transitively on the sphere, in the odd-dimensional case [4]. Namely, the hypothesis of Lemma 5.1 means that the action of H on S^{2n−1} associated to the homomorphism H → Spin(2n) has orbit map H → S^{2n−1} such that π_{2n−1}H_{Q} → π_{2n−1}S_{Q}^{2n−1} ≅ Q is not zero. It follows that H → S^{2n−1} is surjective, in other words that H acts transitively on S^{2n−1}. Then Lemma 5.1 follows from Borel’s classification.

The subgroups we classify next are the simple subgroups H of any simple group G such that the maximal degree of H is at least the secondlargest degree of G. This is closely related to two classifications by Onishchik. First, he classified the simple subgroups H ⊂ G such that dim_{Q} π_{odd}(G/H)_{Q} = 1 ([14], Table 3, p. 185), that is, H kills all but one degree of G; these make up the first part of the list in Lemma 5.2. Next, he classified the simple subgroups H ⊂ G such that d(H) ≥ d(G) − 2 ([22], Table 7.2, p. 195); it turns out that these include all the subgroups on the second part the list in Lemma 5.2.

Lemma 5.2. Let H → G be a nontrivial homomorphism of simply connected simple groups such that the maximal degree of H is at least the second-largest degree of G and is less than the maximal degree of G. Then G/H is isomorphic to one of the following homogeneous spaces:

On the left is the Dynkin index of the homomorphism H → G. On the right are shown the degrees of G not occurring in H, and the degrees of H not occurring in G, in both cases with multiplicities. In the last column is the centralizer of H in G, written modulo finite groups.

1 SU(n)/SU(n − 1)
= S^{2n−1}, n ≥ 3 n S^{1}
1 Sp(2n)/Sp(2n − 2)
= S^{4n−1}, n ≥ 2 2n A_{1}
1 Spin(2n + 1)/Spin(2n)
= S^{2n}, n ≥ 3 2n n 1
1 Spin(2n + 1)/Spin(2n − 1)
= UT(S^{2n}), n ≥ 3  2n  S^{1}
2 Sp(4)/SU(2) = UT(S^{4})  4  S^{1}
10 Sp(4)/SU(2)  4  1
4 SU(3)/SO(3)  3  1
1 Spin(9)/Spin(7) = S^{15}  8  1
1 G_{2}/SU(3) = S^{6}  6 3  1
1 G_{2}/SU(2) = UT(S^{6})  6  A_{1}
3 G_{2}/SU(2)  6  A_{1}
4 G_{2}/SO(3)  6  1
28 G_{2}/SO(3)  6  1
1 F_{4}/Spin(9) = CaP^{2}  12 4  1
1 Spin(2n)/Spin(2n − 2)
= UT(S^{2n−1}), n ≥ 4  n, 2n − 2 n − 1 S^{1}
1 Spin(2n)/Spin(2n − 3), n ≥ 4  n, 2n − 2  A_{1}
1 SU(2n + 1)/Sp(2n), n ≥ 2  3, 5, …, 2n + 1  S^{1}
2 SU(2n + 1)/SO(2n + 1), n ≥ 2 3, 5, …, 2n + 1  1
1 Spin(10)/Spin(7)  5, 8  S^{1}
2 SU(7)/G_{2}  3, 4, 5, 7  1
1 Spin(9)/G_{2}  4, 8  S^{1}
1 Spin(10)/G_{2}  4, 5, 8  A_{1}.

The proof is straightforward for G classical, using the known lowdimensional representations of each simple group, or alternatively the results by Onishchik mentioned above. For G exceptional, more than enough information is provided by Dynkin’s paper on the subgroups of the exceptional groups [8]. For example, Dynkin’s Table 16 classifies the A_{1} subgroups of G_{2}. Notice that the listings for Spin(9)/Spin(7) = S^{15} and Spin(10)/Spin(7) refer to the spin representation of Spin(7); these spaces are different from the spaces Spin(2n + 1)/Spin(2n − 1) = UT(S^{2n}) for n = 4 and Spin(2n)/Spin(2n−3) for n = 5. Also, there are three conjugacy classes of nontrivial homomorphisms SU(2) → Sp(4), where we write V for the standard representation of SU(2): V ⊕ C^{2}, where Sp(4)/SU(2) = Sp(4)/Sp(2) = S^{7}, which has Dynkin index 1; V ⊕ V, where Sp(4)/SU(2) = Spin(5)/Spin(3) = UT(S^{4}), which has Dynkin index 2; and S^{3}V, which has Dynkin index 10.

We now apply Lemma 5.2 to the classification of biquotients. Together with Theorem 4.8, the following theorem will be our most important tool in the classification of biquotients.

Theorem 5.3. Let M = G/H be a simply connected biquotient, written according to Convention 3.4. Let G_{1} be a simple factor of G which contributes only its top degree to M. Then at least one of the following holds:

(1) G_{1} is isomorphic to SU(2), which has degree 2.

(2) G_{1} is a rank-2 group SU(3), Sp(4), or G_{2}, with top degree 3, 4, 6 respectively, and there is a simple factor H_{1} ≅ SU(2) of H which acts nontrivially on G_{1}.

(3) There is a simple factor H_{1} of H such that H_{1} acts nontrivially on exactly one side of G_{1} and G_{1}/H_{1} is one of the homogeneous spaces in the first part of Lemma 5.2’s list.

(4) There are two simple factors H_{1} and H_{2} of H which act nontrivially on the two sides of G_{1} in one of the following ways, up to switching H_{1} and H_{2}: G_{1} = Spin(2n), n ≥ 4, H_{1} is Spin(2n − 2) or Spin(2n − 3), H_{2} is SU(n), or Sp(2a) with n = 2a, or Spin(9) with n = 8; here G_{1} has top degree 2n − 2. Or G_{1} = SU(2n + 1), n ≥ 3, H_{1} is Sp(2n) or SO(2n+1), and H_{2} = SU(2n−1); here G_{1} has top degree 2n + 1. Or H_{1}\ G_{1}/H_{2} is one of G_{2}\ SU(7)/SU(5), G_{2} \ Spin(9)/SU(4), G_{2} \ Spin(9)/Sp(4), or G_{2} \ Spin(10)/SU(5); here G_{1} has top degree 7, 8, 8, 8, respectively.

Proof. If G_{1} = SU(2), we have conclusion (1). So we can assume that G_{1} has rank at least 2. Equivalently, G_{1} has at least two degrees. There must be a simple factor H_{1} of H which kills at least one secondlargest degree of G_{1}. (By Table 4.2, G_{1} has a unique second-largest degree except when G_{1} is Spin(8), which has degrees 2, 4, 4, 6.)

Suppose that H_{1} is isomorphic to G_{1}. If H_{1} acts nontrivially on only one side of G_{1}, then it acts by an isomorphism on that side of G_{1}, and so H_{1} acts transitively on G_{1}, contrary to Convention 3.4. So H_{1} must act by an isomorphism on both sides of G_{1}. Since all automorphisms of G_{1} act as the identity on π_{3}G_{1} = Z, the resulting homomorphism π_{3}(H_{1})_{Q} → π_{3}(G_{1})_{Q} is zero. Also, since H_{1} is acting with finite centralizer on both sides of G_{1}, the rest of H cannot act on G_{1}, and so the whole homomorphism π_{3}H_{Q} → π_{3}(G_{1})_{Q} is zero. That is, G_{1} contributes its degree 2 to M. Since G_{1} is not isomorphic to SU(2), this contradicts our assumption that G_{1} contributes only its top degree to M.

Thus H_{1} is not isomorphic to G_{1}. We know that d(H_{1}) ≤ d(G_{1}) by Lemma 4.3. Suppose that d(H_{1}) = d(G_{1}). By Lemma 4.3, (G_{1}, H_{1}) is one of the pairs (Spin(2n), Spin(2n−1), (SU(2n), Sp(2n)), (Spin(7), G_{2}), (Spin(8), G_{2}), or (E_{6}, F_{4}). In all these cases except (Spin(8), Spin(7)), there is a unique conjugacy class of nontrivial homomorphisms H_{1} → G_{1}; moreover, even in the case (Spin(8), Spin(7)), all nontrivial homomorphisms are equivalent under automorphisms of Spin(8) and so they all give the same homomorphism π_{3}H_{1} → π_{3}G_{1}. Also, the centralizer of H_{1} in G_{1} is finite in all cases. So, in all cases, if H_{1} acts nontrivially on both sides of G_{1}, then no other factor of H acts on G_{1}, and the homomorphism π_{3}H_{Q} → π_{3}(G_{1})_{Q} is zero. That is, G_{1} contributes its degree 2 to M, contrary to our assumption that G_{1} contributes only its top degree to M. Therefore H_{1} must act only on one side of G_{1}. But that implies, in all these cases, that H_{1} kills the top degree of G_{1}. This contradicts our assumption that G_{1} contributes its top degree to M.

So we must have d(H_{1}) < d(G_{1}). Since H_{1} kills at least one secondlargest degree of G_{1}, d(H_{1}) must be at least the second-largest degree of G_{1}. Therefore (G_{1}, H_{1}) must be one of the pairs listed in Lemma 5.2. If H_{1} is isomorphic to SU(2), then G_{1} has rank 2 and we have conclusion (2). We can now assume that H_{1} is not isomorphic to SU(2).

Next, we will show that H_{1} acts nontrivially on only one side of G_{1} in all the remaining cases. Suppose that H_{1} acts nontrivially on both sides of G_{1}. By Lemma 5.2, the centralizer of H_{1} on each side of G_{1} is at most finite by A_{1}. So any simple factor of H other than H_{1} which acts nontrivially on G_{1} is isomorphic to SU(2). Thus H_{1} by itself must kill all the degrees of G_{1} greater than 2 and less than the maximal degree d(G_{1}). This is clearly impossible for the pairs (G_{1}, H_{1}) on the second part of Lemma 5.2’s list, since in these cases G_{1} contains at least one degree in the interval (2, d(G_{1})) with greater multiplicity than H_{1} does.

So (G_{1}, H_{1}) is on the first part of Lemma 5.2’s list. Since H_{1} is not isomorphic to SU(2), the list shows that all nontrivial homomorphisms H_{1} → G_{1} have Dynkin index 1. Since H_{1} acts nontrivially on both sides of G_{1}, it follows that the associated homomorphism π_{3}(H_{1})_{Q} → π_{3}(G_{1})_{Q} is zero. That is, H_{1} does not kill the degree 2 of G_{1}. Since the whole group H does kill the degree 2 of G_{1}, at least one of the two nontrivial homomorphisms H_{1} → G_{1} must have centralizer containing an A_{1} subgroup. By the first part of Lemma 5.2’s list, it follows that the pair (G_{1}, H_{1}) is (Sp(2n), Sp(2n − 2)) for some n ≥ 3.

But for (G_{1}, H_{1}) equal to (Sp(2n), Sp(2n − 2)) with n ≥ 3, there is a unique conjugacy class of nontrivial homomorphisms H_{1} → G_{1}. Since H_{1} acts nontrivially on both sides of G_{1}, it follows that the associated homomorphism π_{∗}(H_{1})_{Q} → π_{∗}(G_{1})_{Q} is zero. In particular, H_{1} does not kill the second-largest degree, 2n − 2, of G_{1}. This is a contradiction. We have thus completed the proof that H_{1} acts nontrivially on only one side of G_{1}.

For spaces G_{1}/H_{1} on the first part of Lemma 5.2’s list, this completes the proof of conclusion (3). It remains to consider spaces G_{1}/H_{1} on the second part of Lemma 5.2’s list. To prove conclusion (4), we need to identify another simple factor of H which acts on G_{1}.

In several cases on the second part of Lemma 5.2’s list, G_{1} is isomorphic to Spin(2n), n ≥ 4. Here H_{1} is either Spin(2n − 2), Spin(2n − 3), or, for n = 5, Spin(7) (with a different homomorphism to Spin(10)) or G_{2}. In all these cases, the degree n of G_{1} is not killed by H_{1}, meaning that π_{2n−1}(H_{1})_{Q} → π_{2n−1}(G_{1})_{Q} is not surjective. So there must be another simple factor H_{2} of H which acts on G_{1} such that π_{2n−1}(H_{2})_{Q} → π_{2n−1}(G_{1})_{Q}/π_{2n−1}(H_{1})_{Q} is not zero. If n is odd, this just means that π_{2n−1}(H_{2})_{Q} → π_{2n−1}(G_{1})_{Q} is not zero. On the other hand, if n is even, then we can always assume, after automorphisms of H_{1} and G_{1}, that H_{1} → G_{1} is the standard inclusion. So, for n odd or even, we can say that

π_{2n−1}(H_{2})_{Q} → π_{2n−1}Spin(2n)_{Q}/π_{2n−1}Spin(2n − 1)_{Q} ≅ Q

is not zero. Here H_{2} cannot be SU(2) since n ≥ 4, so H_{2} must act only on the other side of G_{1} from H_{1}, since the centralizer of H_{1} in G_{1} is at most finite by A_{1}. By Lemma 5.1, the space G_{1}/H_{2} must be one of Spin(2n)/SU(n), Spin(4a)/Sp(2a), Spin(8)/Spin(7), or Spin(16)/Spin(9). If G_{1}/H_{2} = Spin(8)/Spin(7), then we can replace H_{1} by H_{2} and we have conclusion (3); so we can exclude that case. We have thus proved conclusion (4) for G_{1} = Spin(2n).

Next, there are several cases in the second part of Lemma 5.2’s list where G_{1} = SU(2n + 1), n ≥ 2. Here H_{1} is either Sp(2n), SO(2n + 1), or, for n = 3, G_{2}. In all these cases, H_{1} has only even degrees, and in particular it does not kill the degree 2n − 1 of G_{1}. So there must be another simple factor H_{2} of H which kills the degree 2n − 1 of G_{1}. Here H_{2} must act nontrivially only on the other side of G_{1} from H_{1}, because the centralizer of H_{1} in G_{1} is at most S^{1} by finite in these cases. It is easy to read from Table 4.2 that any simple group H_{2} which maps nontrivially to G_{1} = SU(2n + 1), n ≥ 2, and has 2n − 1 as a degree is either SU(2n − 1), SU(2n), or SU(2n + 1). Here H_{2} = SU(2n + 1) is excluded by Convention 3.4, which says that H does not act transitively on G_{1}. If H_{2} = SU(2n), then we can replace H_{1} by H_{2} and we have conclusion (3). The remaining possibility is H_{2} = SU(2n − 1). We have proved conclusion (4) for G_{1} = SU(2n + 1).

The last case, from the second part of Lemma 5.2’s list, is where G_{1} is Spin(9) and H_{1} is the exceptional group G_{2}. Here the degree 4 of G_{1} is not killed by H_{1}, so it must be killed by some other simple factor H_{2} of H. The centralizer of H_{1} in G_{1} is finite by S^{1}, so H_{2} must act nontrivially only on the other side of G_{1} from H_{1}. From Table 4.2, since H_{2} has a degree 4 and maps nontrivially to Spin(9), H_{2} is one of Sp(4), SU(4), Spin(7), Spin(8), or Spin(9). The case H_{2} = Spin(9) is excluded by Convention 3.4, which says that H does not act transitively on G_{1}. If H_{2} is Spin(7) or Spin(8), then we can replace H_{1} by H_{2} and we have conclusion (3). The remaining possibilities for H_{2} are Sp(4) and SU(4). We have proved conclusion (4) for G_{1} = Spin(9). q.e.d.

6. Biquotients which are rational homology spheres

As an application of Theorems 4.8 and 5.3, we now classify all biquotients which are simply connected rational homology spheres. The result seems surprising. In particular, the Gromoll-Meyer exotic 7-sphere which is a biquotient [15] is the only such example in any dimension. As mentioned in the introduction, Theorem 6.1 was found at the same time by Kapovitch and Ziller [17].

Theorem 6.1. Any biquotient which is a simply connected rational homology sphere is either a homogeneous manifold, the Gromoll-Meyer exotic 7-sphere which is a biquotient Sp(4)/SU(2), or a certain 4-connected 11-manifold with the integral homology groups of UT(S^{6}) which is a biquotient G_{2}/SU(2).

The homogeneous manifolds which are simply connected rational homology spheres are the sphere S^{n}, the unit tangent bundle UT(S^{2n}), the Wu 5-manifold SU(3)/SO(3) [7], the Berger 7-manifold Sp(4)/SU(2) with π_{3}M isomorphic to Z/10 [3], the 11-manifold G_{2}/SU(2) with π_{3}M isomorphic to Z/3, and the two 11-manifolds G_{2}/SO(3) with π_{3}M isomorphic to Z/4 or Z/28.

The homogeneous spaces in Theorem 6.1 were classified by Onishchik ([14], Table 3, p. 185). The nontrivial biquotient G_{2}/SU(2) was first discovered by Eschenburg ([10], pp. 166-170). Kapovitch and Ziller have shown that this biquotient G_{2}/SU(2) is diffeomorphic to the connected sum of UT(S^{6}) with some homotopy 11-sphere [17]. It is not known whether G_{2}/SU(2) is actually diffeomorphic to UT(S^{6}); one hopes for a negative answer, which would be more interesting.

Before starting the proof of Theorem 6.1, we assemble some elementary facts about biquotients Sp(4)/H. In the following lemma, we consider free actions of a group H on Sp(4) given by a homomorphism H → Sp(4)^{2}/Z(Sp(4)).

Lemma 6.2.

(1) There is no free action of SO(3) on Sp(4). Any free action of SU(2) on Sp(4) is either trivial on one side of Sp(4), so that Sp(4)/SU(2) is S^{7}, UT(S^{4}), or the Berger 7-manifold, or given by the homomorphisms (V ⊕C^{2}, V^{⊕2}) up to switching the two sides of Sp(4), so that Sp(4)/SU(2) is the Gromoll-Meyer exotic 7-sphere [15]. Here V denotes the standard representation of SU(2).

(2) Any free action of SU(2)^{2} on Sp(4) is given, up to switching the two SU(2) factors and switching the two sides of Sp(4), by the homomorphisms (V_{1} ⊕ V_{2}, C^{4}) or (V_{1} ⊕ C^{2}, V_{2}^{⊕2}). Here V_{1} and V_{2} are the standard representations of the two SU(2) factors. In both cases, the quotient Sp(4)/SU(2)^{2} is diffeomorphic to S^{4}.

Proof. We only prove (1) here, the calculation for (2) being similar. There are three conjugacy classes of nontrivial homomorphisms SU(2) → Sp(4), V ⊕ C^{2}, V^{⊕2}, and S^{3}V, by Lemma 5.2. If SU(2) acts trivially on one side of Sp(4), then Sp(4)/SU(2) is one of the three homogeneous spaces mentioned in (1). So we can assume that SU(2) acts nontrivially on both sides of Sp(4). Let H be SU(2) or SU(2)/{±1} ≅ SO(3). The group H acts freely on Sp(4) if and only if the two images of each nontrivial element of H in Sp(4) are not conjugate. In particular, the two homomorphisms SU(2) → Sp(4) must be non-conjugate. If one homomorphism is V ⊕ C^{2} and the other is V^{⊕2}, then Gromoll and Meyer showed that SU(2) acts freely on Sp(4) and that the quotient manifold is an exotic 7-sphere [15]. Otherwise, SU(2) must act by S^{3}V on one side and either V ⊕ C^{2} or V^{⊕2} on the other. Then neither SU(2) nor SU(2)/{±1} acts freely on Sp(4), since the images of the diagonal matrix (ζ_{3}, ζ_{3}^{−1}) in SU(2) under S^{3}V and V ⊕ C^{2} are conjugate, and likewise the images of the diagonal matrix (ζ_{4}, ζ_{4}^{−1}) in SU(2) under S^{3}V and V^{⊕2} are conjugate. (Here ζ_{n} denotes a primitive nth root of unity.) This proves (1). q.e.d.

Proof of Theorem 6.1. We write M = G/H according to Convention 3.4. Since M is a rational homology sphere, the odd-dimensional rational homotopy of M has dimension 1. In more detail, S^{2n−1} has 1dimensional rational homotopy in dimension 2n − 1 and zero otherwise, while S^{2n} has 1-dimensional rational homotopy in dimensions 2n and 4n − 1, and zero otherwise. Since each simple factor of G contributes at least dimension 1 to π_{odd}M_{Q} by Theorem 4.8, G must be simple. This already makes the situation much more understandable; it would not be clear at all without the analysis leading to Theorem 4.8.

If the simply connected rational homology sphere M has dimension r at most 4, then it is automatically a homotopy sphere. It is therefore not surprising to find that M is diffeomorphic to S^{r} for r ≤ 4. First, for r = 2 we can just use that every homotopy 2-sphere is diffeomorphic to S^{2}. For r ≥ 3, we have π_{2}M_{Q} = 0, and so H has finite fundamental group; equivalently, H is semisimple. For r = 3, π_{3}M ≅ Z and π_{4}M_{Q} = 0, which implies that G has one more simple factor than H does. Since G is simple, H = 1. Since M is a homotopy 3-sphere, G has degree 2 only, and so G is SU(2). Thus M is diffeomorphic to SU(2), that is, to S^{3}.

For r = 4, π_{2}M = π_{3}M = 0 and π_{4}M ≅ Z, which implies that H is simply connected and has one more simple factor than G does. Since G is simple, H has two simple factors. By the rational homotopy groups of M, G must contribute degree 4 to M while H contributes degree 2, and nothing else. By Theorems 4.8 and 5.3, there is a simple factor H_{1} of H such that either G/H_{1} is a homogeneous space diffeomorphic to S^{7}, or G/H_{1} is Spin(8)/G_{2} = S^{7} × S^{7}, or (G, H_{1}) is (Sp(4), SU(2)). Let H_{2} denote the other simple factor of H. If G/H_{1} = S^{7}, then H_{2} has degree 2 only, so H_{2} is isomorphic to SU(2). In this case M is the quotient of S^{7} by SU(2) acting freely by a homomorphism SU(2) → O(8). Such a homomorphism is unique up to conjugacy, and so M is the standard quotient S^{4}. Next, if G/H_{1} is Spin(8)/G_{2} = S^{7}×S^{7}, then H_{2} has degrees 2 and 4, and so H_{2} is isomorphic to Sp(4). But there is in fact no free isometric action of Sp(4) on S^{7}×S^{7}, because the restriction of any action of Sp(4) on S^{7} to the subgroup SU(2) = Sp(2) ⊂ Sp(4) has a fixed point. (Either the associated 8-dimensional complex representation of Sp(4) has a trivial summand, or it is the sum of two copies of the standard representation of Sp(4) and hence restricts on SU(2) to V^{⊕2} ⊕ C^{4}.)

For r = 4, it remains to consider the case (G, H_{1}) = (Sp(4), SU(2)). Here H_{2} has degree 2 only, and so H_{2} is isomorphic to SU(2). Thus M is a biquotient Sp(4)/SU(2)^{2}. By Lemma 6.2, M is diffeomorphic to S^{4}.

We proceed to the case r ≥ 5. As mentioned earlier, we have π_{2}M_{Q} = 0, and so H is semisimple. Since π_{3}M_{Q} = π_{4}M_{Q} = 0, H has the same number of simple factors as G. Since G is simple, so is H.

One of the cases (1) to (4) in Theorem 4.8 must hold. Here (4) is excluded since G is simple and H acts freely on G. In case (3), M is the homogeneous space Spin(2n)/Spin(2n − 1) = S^{2n−1}. In case (2), M is one of the homogeneous spaces SU(2a)/Sp(2a) with a ≥ 2, Spin(7)/G_{2} = S^{7}, Spin(8)/G_{2} = S^{7} × S^{7}, or E_{6}/F_{4}. Since M is a rational homology sphere, considering the degrees of these homogeneous spaces shows that M is either SU(4)/Sp(4) = Spin(6)/Spin(5) = S^{5} or Spin(7)/G_{2} = S^{7}.

There remains case (1) of Theorem 4.8, where G contributes its top degree to M. In this case, one of conclusions (1) to (4) in Theorem 5.3 must hold. Here case (4) of Theorem 5.3 is excluded since H is simple. In case (3), M is one of the homogeneous spaces listed in the first part of Lemma 5.2. Since M is a rational homology sphere, the possibilities are as listed in Theorem 6.1.

Case (1), G = SU(2), of Theorem 5.3 is excluded because we are considering biquotients M of dimension r ≥ 5. So there remains only case (2). That is, H_{1} is SU(2), H is either H_{1} = SU(2) or H_{1}/{±1} ≅ SO(3), and M is a biquotient SU(3)/H, Sp(4)/H, or G_{2}/H, of dimension 5, 7, or 11, respectively. The homogeneous spaces of this type are listed in Lemma 5.2. So we can assume that H_{1} acts nontrivially on both sides of G. Biquotients of this type (a rank-2 group divided by a rank-1 group) were classified by Eschenburg ([10], pp. 166–170), but I will give my own proof since Eschenburg’s paper is not widely available.

First, suppose G = SU(3). Then there are two conjugacy classes of nontrivial homomorphisms SU(2) → SU(3), V ⊕ C and S^{2}V, where V denotes the standard representation of SU(2). Since H (which is SU(2) or SO(3)) acts freely on G, the two images in SU(3) of any nontrivial element of H are not conjugate in SU(3). So SU(2) must act by V ⊕ C on one side of SU(3) and by S^{2}V on the other. But, if we write ζ_{a} for a primitive ath root of unity, the diagonal matrix (ζ_{3}, ζ_{3}^{−1}) in SU(2) has images under both homomorphisms to SU(3) which are conjugate to (ζ_{3}, 1, ζ_{3}^{−1}). So neither SU(2) nor SU(2)/{±1} can be acting freely. Thus, the case G = SU(3) does not occur.

The case G = Sp(4) is covered by Lemma 6.2. Since SU(2) acts nontrivially on both sides of Sp(4), M is the Gromoll-Meyer exotic 7sphere.

The last case to consider is where G is the exceptional group G_{2}. Here, as described in Lemma 5.2, there are 4 conjugacy classes of nontrivial homomorphisms SU(2) → G_{2}, which we identify by their Dynkin index, 1, 3, 4, or 28. Using these Dynkin indices, we compute that the composed representation SU(2) → G_{2} → U(7) must be: W_{1} = V^{⊕2}⊕C^{3}, W_{3} = S^{2}V ⊕ V^{⊕2}, W_{4} = (S^{2}V)^{⊕2} ⊕ C, and W_{28} = S^{6}V. In particular, the homomorphisms W_{4} and W_{28} from SU(2) to G_{2} are trivial on {±1} in SU(2), and the other two are not.

We are assuming that SU(2) acts nontrivially on both sides of G_{2}. Since the action is free for either SU(2) or SU(2)/{±1}, the two homomorphisms SU(2) → G_{2} must be non-conjugate. There are now 6 cases to consider, corresponding to the 6 unordered pairs of distinct homomorphisms W_{i}. Notice that the action of SU(2) on G_{2} is trivial on {±1} if and only if the two homomorphisms send −1 to the same element of the center of G_{2}. The center of G_{2} is trivial, so the two homomorphisms must both be trivial on −1. That is, SU(2) acts on G_{2} through its quotient SO(3) if and only if the two homomorphisms are (W_{4}, W_{28}).

Given that SU(2) acts on G_{2} by W_{i} on one side and W_{j} on the other, the action of SU(2) is free if and only if every nontrivial element of SU(2) has non-conjugate images in G_{2} under the two homomorphisms. Every element of SU(2) is conjugate to a diagonal element (x, x^{−1}) with x ∈ S^{1}, so it suffices to consider those elements. Furthermore, it is convenient to observe that two elements of G_{2} are conjugate if and only if their images in U(7) are conjugate.

Using this, we can check whether each pair (W_{i}, W_{j}) of homomorphisms SU(2) → G_{2} gives a free action of SU(2) on G_{2} or not. Here the image of x ∈ S^{1} ⊂ SU(2) in U(7) under the homomorphism W_{i} is conjugate to the diagonal matrix:

W_{1} : (x, x^{−1}, x, x^{−1}, 1, 1, 1)
W_{3} : (x, x^{−1}, x, x^{−1}, x^{2}, 1, x^{−2})
W_{4} : (x^{2}, x^{−2}, x^{2}, x^{−2}, 1, 1, 1)
W_{28} : (x^{−6}, x^{−4}, x^{−2}, 1, x^{2}, x^{4}, x^{6}).

The result is that 5 of the 6 unordered pairs (W_{i}, W_{j}) do not give free actions of SU(2) (or SO(3), in the case (W_{4}, W_{28})) on G_{2}. Indeed, the following nontrivial elements x ∈ S^{1} have conjugate images (in U(7), hence in G_{2}) under the two representations.

(W_{1}, W_{3}) : x = −1
(W_{1}, W_{4}) : x = ζ_{3}
(W_{1}, W_{28}) : x = ζ_{3}
(W_{3}, W_{28}) : x = ζ_{5}
(W_{4}, W_{28}) : x = ζ_{3}

If SU(2) acts on G_{2} by (W_{3}, W_{4}), however, then we compute from the above formulas that the two images of any nontrivial element x ∈ S^{1} ⊂ SU(2) are not conjugate in U(7), and hence not conjugate in G_{2}. So this is a free action of SU(2) on G_{2}. Since the Dynkin indices 3 and 4 differ by 1, the resulting biquotient M = G_{2}/SU(2) has π_{3}M = 0. Using the known homology of G_{2}, we compute that the 11-manifold M is 4-connected and has the integral homology groups of UT(S^{6}).

Thus, the only biquotient G_{2}/H with H isomorphic to SU(2) or SO(3) and H acting nontrivially on both sides is the 4-connected 11manifold G_{2}/SU(2), with SU(2) acting by (W_{3}, W_{4}). q.e.d.

7. Three Cheeger manifolds which are not homotopy equivalent to biquotients

We now return to the proof of Theorem 2.1. Roughly in order of increasing difficulty, we show in this section that the 16-manifolds CaP^{2}#CaP^{2}, CP^{8}#CaP^{2}, and HP^{4}#CaP^{2} are not homotopy equivalent to biquotients.

Suppose that the 16-manifold CaP^{2}#CaP^{2} is homotopy equivalent to a biquotient M = G/H. As throughout the paper, we assume that M is written as a biquotient G/H which satisfies Convention 3.4. Here H^{∗}(M, Z) ≅ Z[x, y]/(xy = 0, x^{2} = y^{2}), |x| = |y| = 8. This is a complete intersection ring with 2 generators in degree 8 and 2 relations in degree 16. It follows that the rational homotopy groups of M are isomorphic to Q^{2} in dimensions 8 and 15, and otherwise 0 (the same as for S^{8} × S^{8}). So the map π_{2d−1}H_{Q} → π_{2d−1}G_{Q} must have 2-dimensional cokernel for d = 8, 2-dimensional kernel for d = 4, and otherwise is an isomorphism. Equivalently, we say that G contributes two degrees 8 to M, while H contributes two degrees 4, and nothing else.

Each simple factor of G contributes at least one degree to M, by Theorem 4.8, so G has at most two simple factors. Suppose first that G is simple. We know that G contributes two degrees 8 to M, and nothing else. One of the four cases in Theorem 4.8 must hold. Here (1) cannot hold: if G contributes its maximal degree to M, that would have to be 8, but the maximal degree of each simple Lie group occurs only with multiplicity 1. The remaining cases of the theorem are incompatible with the fact that G contributes only degree 8 to M. Thus we have a contradiction from the assumption that G is simple.

So G has two simple factors, G = G_{1}×G_{2}. (Here G_{2} does not denote the exceptional group G_{2}.) Each factor must contribute one degree 8 to M, and nothing else. We can apply Theorem 4.8 to each factor G_{i}. From the degrees, it is clear that only cases (1) and (3) can occur. That is, for 1 ≤ i ≤ 2, either G_{i} contributes its maximal degree to M, which must be 8, or else G_{i} = Spin(16) and there is a simple factor H_{i} ≅ Spin(15) of H which acts nontrivially on exactly one side of G_{i}, with G_{i}/H_{i} = S^{15}. If G_{i} contributes its maximal degree to M, we can apply Theorem 5.3 to G_{i}. Cases (1) and (2) cannot arise, since the maximal degree of G_{i} is 8. Therefore, by cases (3) and (4), there is a simple factor H_{i} of H which acts nontrivially on exactly one side of G_{i} such that G_{i}/H_{i} is one of the homogeneous spaces Spin(9)/Spin(8) = S^{8}, SU(8)/SU(7) = Sp(8)/Sp(6) = Spin(9)/Spin(7) = S^{15}, Spin(9)/Spin(7) = UT(S^{8}), Spin(10)/Spin(8), Spin(10)/Spin(7), Spin(9)/G_{2}, or Spin(10)/G_{2}.

Since neither G nor H contributes any degree 2 to M, and each simple group has exactly one degree 2 (that is, every simple group has π_{3} = Z), H must have the same number of simple factors as G. Thus H has two simple factors.

Let G_{1} and G_{2} denote the two simple factors of G. We know that there is a simple factor H_{1} of H such that H_{1} acts nontrivially on exactly one side of G_{1}, with G_{1}/H_{1} equal to either Spin(16)/Spin(15) = S^{15} or one of the other homogeneous spaces listed above. We also know the analogous statement for G_{2}, but a priori it is possible that the same simple factor H_{1} of H plays the same role for both G_{1} and G_{2}. But then G = G_{1} × G_{2} has two degrees 14 (if G_{1} and G_{2} are Spin(16)) or 7 (if G_{1} and G_{2} are SU(8)) or 6 (if G_{1} and G_{2} are Spin(9) or Spin(10)), both of which must be killed by H, whereas the group H_{1} has only one degree 14 or 6 or 7. So we can order the two simple factors of H in such a way that H_{i} kills the relevant degree of G_{i}, both for i = 1 and for i = 2. By the proof of Theorem 5.3, it follows that H_{i} acts nontrivially on exactly one side of G_{i}, with G_{i}/H_{i} equal to one of the above homogeneous spaces, for i = 1 and for i = 2.

Since M has dimension only 16, both G_{1}/H_{1} and G_{2}/H_{2} must be the homogeneous space Spin(9)/Spin(8) = S^{8}. If H_{1} acts trivially on G_{2}, or also if H_{2} acts trivially on G_{1}, then M is an S^{8}-bundle over S^{8} and hence has signature zero, contradicting the fact that M is homotopy equivalent to CaP^{2}#CaP^{2}. So H_{1} acts nontrivially on G_{2} and H_{2} acts nontrivially on G_{1}. Since H_{i} has finite centralizer in G_{i} for i = 1, 2, the action of H_{1} on G_{2} must be given by a nontrivial homomorphism H_{1} → G_{2} on the other side of G_{2} from H_{2}, and likewise for the action of H_{2} on G_{1}. Any nontrivial homomorphism Spin(8) → Spin(9) has Dynkin index 1. So the homomorphism π_{3}H → π_{3}G, Z^{2} → Z^{2}, is given by a 2 × 2 matrix with both rows equal to (1, −1) or (−1, 1). Such a matrix has zero determinant, and so π_{3}M_{Q} is not zero. Again, this contradicts the fact that M is homotopy equivalent to CaP^{2}#CaP^{2}. This completes the proof that CaP^{2}#CaP^{2} is not homotopy equivalent to a biquotient. In fact, the proof shows that CaP^{2}#CaP^{2} is not even rationally homotopy equivalent to a biquotient.

We now prove that CP^{8}#CaP^{2} is not homotopy equivalent to a biquotient M = G/K. Since G is simply connected, the boundary map Z ≅ π_{2}M → π_{1}K in the long exact sequence is an isomorphism.

So, if we let H be the commutator subgroup of K, then H is simply connected and K/H is isomorphic to S^{1}. Let N be the 17-manifold G/H. Since N is the natural S^{1}-bundle over M ≃ CP^{8}#CaP^{2}, we compute that N has the integral cohomology ring of S^{8} × S^{9}. To go further, we use Wu’s theorem that the Stiefel-Whitney classes of the tangent bundle of M are invariants of the homotopy type of M ([19], Theorem 11.14). Because the 8-sphere in the Cayley plane has selfintersection 1, the Stiefel-Whitney class w_{8}(CaP^{2}) ∈ H^{8}(CaP^{2}, F_{2}) is not zero. Therefore w_{8}(CP^{8}#CaP^{2}) is not in the subgroup (H^{2})^{4} of H^{8}. By Wu’s theorem, the same holds for the manifold M which is homotopy equivalent to CP^{8}#CaP^{2}. Since N is an S^{1}-bundle over M, it follows that w_{8}(N) is not zero. We will use this later.

The cohomology ring of M is a complete intersection ring. The degrees of the generators and relations determine the rational homotopy groups of M, and hence of N. The result is that G contributes degrees 5 and 8 to N, while H contributes 4, and nothing else. Since every simple factor of G contributes at least one degree to N, G has at most 2 simple factors. Also, since neither G nor H contributes degree 2, G and H have the same number of simple factors.

Suppose first that G is simple. It follows that H is simple, too. We can apply Theorem 4.8, and cases (2), (3), (4) are excluded since G contributes degrees 5 and 8 to N and no more. Therefore case (1) must hold; that is, G contributes its maximal degree to N. So G has maximal degree 8. There is no simple group whose highest two degrees are 5 and 8, so the second-largest degree of G must be killed by H. Since G has exactly one degree 8, H must have maximal degree less than 8. So, regardless of how H acts on G, (G, H) must be one of the pairs in Lemma 5.2. From the list there, since we know that G adds degrees 5 and 8 to M and H adds degree 4 and no more, we must have G = Spin(10) and H = Spin(8). All nontrivial homomorphisms H → G have Dynkin index 1. So if H acts nontrivially on both sides of G, then G would contribute degree 2 to M, a contradiction. So H acts on only one side of G and G/H is isomorphic to the homogeneous space Spin(10)/Spin(8) = UT(S^{9}).

Indeed, there is a free S^{1}-action on UT(S^{9}), with quotient the 8dimensional complex quadric Q_{C}^{8}. This quadric has the rational homotopy type of CP^{8}#CaP^{2}. But no S^{1}-quotient of N = UT(S^{9}) can have the homotopy type of CP^{8}#CaP^{2}. The point is that N is an S^{8}-bundle over S^{9}. It follows that there is a sphere S^{8} in N which represents a generator of H_{8}(N, F_{2}) and which has trivial normal bundle. Therefore w_{8}(N)|S^{8} = 0, and hence w_{8}(N) = 0, contradicting our earlier calculation that the Stiefel-Whitney class w_{8}(N) is not zero.

So G must have two simple factors. Since G and H have the same number of simple factors, H also has two simple factors. Each simple factor of G must contribute exactly one degree to N; we can assume that G_{1} contributes degree 8 and G_{2} contributes degree 5. By Theorems 4.8 and 5.3, there is a simple factor H_{1} acting on exactly one side of G_{1} such that G_{1}/H_{1} is one of the homogeneous spaces Spin(9)/Spin(8) = S^{8}, Spin(16)/Spin(15) = SU(8)/SU(7) = Sp(8)/Sp(6) = Spin(9)/Spin(7) = S^{15}, Spin(9)/Spin(7) = UT(S^{8}), Spin(10)/Spin(8), Spin(10)/Spin(7), Spin(9)/G_{2}, or Spin(10)/G_{2}. Also, by the same theorems, there is a simple factor H_{2} acting on exactly one side of G_{2} such that G_{2}/H_{2} is one of the homogeneous spaces Spin(10)/Spin(9) = SU(5)/SU(4) = S^{9} or SU(6)/Sp(6).

If H_{1} and H_{2} are the same factor of H, this factor must be isomorphic to Sp(6), and N is a biquotient of the form (Sp(8)×SU(6))/(Sp(6)×X). Here X must be a simple group with degrees 2, 3, 4, 4, 6, but there is no such group. So H_{1} and H_{2} are the two different simple factors of H.

From the degrees of N (or just by its dimension, which is only 17), the homogeneous space G_{1}/H_{1} must be Spin(9)/Spin(8) = S^{8} and the homogeneous space G_{2}/H_{2} must be either Spin(10)/Spin(9) = S^{9} or SU(5)/SU(4) = S^{9}. The 17-manifold N is not an S^{8}-bundle over S^{9}, because we know that w_{8}(N) is not zero. So H_{1} = Spin(8) must act nontrivially on the second factor G_{2} of G. It follows that G_{2}/H_{2} is Spin(10)/Spin(9) = S^{9}, not SU(5)/SU(4) = S^{9}. Furthermore, H_{2} = Spin(9) must act trivially on G_{1} = Spin(9), by Lemma 4.1. Thus N is a biquotient of the form (Spin(9) × Spin(10))/(Spin(8) × Spin(9)) with Spin(9) acting trivially on Spin(9) and Spin(8) acting nontrivially on Spin(10). That is, N is a nontrivial S^{9}-bundle over S^{8}.

There are three conjugacy classes of nontrivial homomorphisms from Spin(8) to Spin(10). They have the form W ⊕ R^{2}, where W is either the standard 8-dimensional real representation V of Spin(8) or one of the two spin representations S^{−} and S^{+}. Thus N is the S^{9}- bundle S(W ⊕ R^{2}) over S^{8}, where W is the real vector bundle over Spin(9)/Spin(8) = S^{8} corresponding to the representation W of Spin(8). If W is the standard representation V of Spin(8), then the corresponding vector bundle on S^{8} is the tangent bundle, which has zero Stiefel-Whitney classes. It follows that the sphere bundle S(V ⊕ R^{2}) has zero Stiefel-Whitney classes, contrary to what we know about N. So W must be one of the two spin representations of Spin(8). Without loss of generality, we can assume that W = S^{−}; if instead W = S^{+}, we can apply an automorphism of order 2 of Spin(8) which switches the isomorphism classes of the representations S^{−} and S^{+} and does not change the isomorphism class of the representation V and hence of the standard inclusion Spin(8) → Spin(9). Once that is done, N is the S^{9}-bundle S(S^{−} ⊕ R^{2}) over S^{8}.

Finally, we need to analyze the free S^{1}-action on N that gives M = N/S^{1}. By computing the centralizers of the homomorphisms that define N, we see that the S^{1}-action on N is defined by a homomorphism S^{1} → Spin(9) together with a homomorphism from S^{1} to S^{1}, the identity component of the centralizer of Spin(8) in Spin(10). The homomorphism S^{1} → Spin(9) defines an action of S^{1} on S^{8} and also on the Spin(9)- equivariant vector bundle S^{−} over S^{9}, while the homomorphism S^{1} → S^{1}, of the form z ↦ z^{b} for some integer b, defines the action of S^{1} on the trivial bundle R^{2} over S^{8}. Since S^{1} is acting freely on N = S(S^{−} ⊕ R^{2}), it must act freely on both S(S^{−}) = S^{15} and on S(R^{2}) = S^{1} ×S^{8}. Since the S^{1}-action on S^{8} must have a fixed point, the freeness of the S^{1}-action on S(R^{2}) means that b = ±1. We compute that there is a unique conjugacy class of homomorphisms S^{1} → Spin(9) which give a free action of S^{1} on S(S^{−}) = S^{15}, namely the subgroup S^{1} = Spin(2) ⊂ Spin(9). We can assume that b = 1, since changing b = −1 to b = 1 clearly does not change the diffeomorphism class of the quotient manifold M = N/S^{1}. Thus we have uniquely described the biquotient M.

But the biquotient M we have described is exactly the one which we proved to be diffeomorphic to CP^{8}#−CaP^{2} in Section 2. In particular, it is not homotopy equivalent to CP^{8}#CaP^{2}. This completes the proof that CP^{8}#CaP^{2} is not homotopy equivalent to a biquotient.

We now show that HP^{4}#CaP^{2} is not homotopy equivalent to a biquotient M = G/H. Since M is 2-connected, H is simply connected, by Lemma 3.1. The ring H^{∗}(M, Z) is a complete intersection, and so the degrees of its generators and relations determine the rational homotopy groups of M. The result is that G contributes degrees 6 and 8 to M, while H contributes degrees 2 and 4, and no others. Since every simple factor of G contributes at least one degree to M, G has at most 2 simple factors. Since H contributes exactly one degree 2, H has one more simple factor than G has.

Suppose first that G is simple. We can apply Theorem 4.8, and cases (2), (3), (4) are excluded because G contributes degrees 6 and 8 to M and no more. Therefore case (1) must hold; that is, G contributes its maximal degree to M. So G has maximal degree 8. By Table 4.2, G must be one of Sp(8), Spin(9), Spin(10), or SU(8), which have degrees respectively 2, 4, 6, 8, 2, 4, 6, 8, 2, 4, 5, 6, 8, or 2, 3, 4, 5, 6, 7, 8. It follows that H has, correspondingly, degrees 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 5, or 2, 2, 3, 4, 4, 5, 7. But there is no semisimple group H with the last two sets of degrees: one simple factor of H would have to have maximal degree equal to an odd number d (5 or 7), hence would be isomorphic to SU(d), and hence would have all degrees from 2 to d, which is not the case here. Therefore G is isomorphic to Sp(8) or Spin(9), and H has degrees 2, 2, 4, 4. Since H is simply connected, Table 4.2 shows that H is isomorphic to Sp(4)^{2}.

We compute that there is no free action of Sp(4)^{2} on Spin(9), and that the only free action of Sp(4)^{2} on Sp(8) is the one-sided action given by the standard inclusion Sp(4)^{2} ⊂ Sp(8). So M is the homogeneous space Sp(8)/Sp(4)^{2}. This homogeneous space has the rational homotopy type of HP^{4}#CaP^{2}, but not the homotopy type, because we compute that w_{4}(Sp(8)/Sp(4)^{2}) = 0 whereas w_{4}(HP^{4}#CaP^{2}) ≠ 0, since w_{4}(HP^{4}) ≠ 0. To compute these Stiefel-Whitney classes of the homogeneous spaces Sp(8)/Sp(4)^{2} and HP^{4}, we can use Singhof’s approach, which works more generally for biquotients [25]. Namely, say for X = Sp(8)/Sp(4)^{2}, the tangent bundle is

T X = sp(8) − sp(4)_{1} − sp(4)_{2}

in the Grothendieck group of real vector bundles on X. The groups H^{i}(BSp(2n), F_{2}) are zero for 0 < i < 4, and so

w_{4}X = w_{4}(sp(8)) − w_{4}(sp(4)_{1}) − w_{4}(sp(4)_{2}).

This is zero because w_{4}(sp(4)_{1})+w_{4}(sp(4)_{2}) is clearly some F_{2}-multiple of the sum of the generators of H^{4}(BSp(4)_{1}, F_{2}) and of H^{4}(BSp(4)_{2}, F_{2}), while the generator of H^{4}(BSp(8), F_{2}) pulls back to the sum of the generators of H^{4}(BSp(4)_{1}, F_{2}) and H^{4}(BSp(4)_{2}, F_{2}) and also pulls back to zero in H^{4}(X, F_{2}).

Therefore G is not simple. It must have two simple factors, each of which contributes exactly one degree to M. We know that H has one more simple factor than G, so H has three simple factors. We can write G = G_{1} × G_{2} where G_{1} contributes degree 6 to M and G_{2} contributes degree 8 to M, and nothing else. By Theorems 4.8 and 5.3 applied to G_{1}, there is a simple factor H_{1} of H such that either G_{1} is the exceptional group G_{2}, H_{1} is SU(2), and H_{1} acts nontrivially on G_{1}, or H_{1} acts nontrivially on exactly one side of G_{1} and G_{1}/H_{1} is isomorphic to one of the homogeneous spaces Spin(7)/Spin(6) = G_{2}/SU(3) = S^{6}, Spin(12)/Spin(11) = SU(6)/SU(5) = Sp(6)/Sp(4) = S^{11}, Spin(7)/Spin(5) = UT(S^{6}), Spin(8)/Spin(6), or Spin(8)/Spin(5). Likewise, by Theorems 4.8 and 5.3 applied to the second factor G_{2}, there is a simple factor H_{2} of H which acts nontrivially on exactly one side of G_{2} with G_{2}/H_{2} isomorphic to one of the homogeneous spaces Spin(9)/Spin(8) = S^{8}, Spin(16)/Spin(15) = SU(8)/SU(7) = Sp(8)/Sp(6) = Spin(9)/Spin(7) = S^{15}, Spin(9)/Spin(7) = UT(S^{8}), Spin(10)/Spin(8), Spin(10)/Spin(7), Spin(9)/G_{2}, or Spin(10)/G_{2}.

We see that in all the above cases G_{1}/H_{1} and G_{2}/H_{2}, H_{1} is never isomorphic to H_{2}, so in particular H_{1} is not the same simple factor of H as H_{2}. So, given G_{1}/H_{1} and G_{2}/H_{2}, the known degrees of M determine the degrees of the remaining simple factor of H, H_{3}. Let us say that a given set of factors of G and H, for example G_{1} and H_{1}, adds a degrees d to M if the integer a is the number of degrees d in the given factors of G minus the number of degrees d in the given factors of H. From the above list, G_{1}/H_{1} always adds exactly one degree 6 to M, while G_{2}/H_{2} adds none; so, by the known degrees of M, H_{3} has no degree 6. The only simple group with two degrees 4 is Spin(8), which also has a degree 6; so H_{3} has at most one degree 4.

Also, from the above list, G_{2}/H_{2} adds no degree 3 to M, so the cases where G_{1}/H_{1} adds −1 degree 3 to M cannot occur (otherwise H_{3} would have −1 degrees 3). Thus G_{1}/H_{1} is not Spin(7)/Spin(6) = S^{6}, G_{2}/SU(3) = S^{6}, or Spin(8)/Spin(6) = UT(S^{7}). Thus G_{1}/H_{1} and G_{2}/H_{2} both add zero degrees 3 to M, and so H_{3} has no degree 3. Next, we observe that G_{1}/H_{1} adds ≥ 0 degrees 4 to M, so the cases where G_{2}/H_{2} adds one degree 4 to M cannot occur (otherwise H_{3} would have at least two degrees 4). Thus G_{2}/H_{2} is not Spin(9)/G_{2} or Spin(10)/G_{2}. Next, G_{1}/H_{1} adds zero degrees 5 to M, so in the cases where G_{2}/H_{2} adds degree 5 to M, H_{3} must have a degree 5. Since H_{3} has no degree 6, it follows that H_{3} is isomorphic to SU(5), contradicting the fact that H_{3} has no degree 3. Thus G_{2}/H_{2} is not Spin(10)/Spin(8) or Spin(10)/Spin(7).

Finally, if G_{1}/H_{1} is Spin(8)/Spin(5) = Spin(8)/Sp(4), then G_{1}/H_{1} adds one degree 4 to M, and so, since H_{3} has at most one degree 4, G_{2}/H_{2} must add −1 degrees 4 to M; that is, G_{2}/H_{2} is Spin(9)/Spin(8) = S^{8}. Then H_{3} has degrees 2, 4 and so H_{3} is isomorphic to Sp(4). Here M is a biquotient (Spin(8) × Spin(9))/(Sp(4) × Spin(8) × Sp(4)). Since H_{2} = Spin(8) has finite centralizer in G_{2} = Spin(9), the two Sp(4) factors of H can only act on Spin(9) on the other side from Spin(8); so Sp(4)^{2} acts on S^{8}. Looking at the low-dimensional orthogonal representations of Sp(4)^{2} shows that this action must have a fixed point. Since H acts freely on G, it follows that there is a subgroup of H isomorphic to Sp(4)^{2} which acts freely on Spin(8). We compute, however, that there is no free action of Sp(4)^{2} on Spin(8). This contradiction shows that G_{1}/H_{1} is not Spin(8)/Spin(5) = Spin(8)/Sp(4).

The only remaining possibilities for G_{1}/H_{1} are those which add degree 6 to M and nothing else: a homogeneous space Spin(12)/Spin(11) = SU(6)/SU(5) = Sp(6)/Sp(4) = S^{11} or Spin(7)/Spin(5) = UT(S^{6}) or (G_{1}, H_{1}) = (G_{2}, A_{1}) (the exceptional group G_{2}). Also, G_{2}/H_{2} is either Spin(9)/Spin(8) = S^{8}, which adds one degree 8 and subtracts one degree 4 from M, or else a homogeneous space which adds degree 8 to M and nothing else: Spin(16)/Spin(15) = SU(8)/SU(7) = Sp(8)/Sp(6) = Spin(9)/Spin(7) = S^{15} or Spin(9)/Spin(7) = UT(S^{8}). From the known degrees of M, it follows that H_{3} has only degree 2 if G_{2}/H_{2} is S^{8} and has degrees 2, 4 otherwise. Therefore H_{3} is isomorphic to SU(2) if G_{2}/H_{2} is S^{8} and to Sp(4) otherwise.

Suppose that H_{3} is isomorphic to Sp(4); we will derive a contradiction. First, we can easily exclude the possibility that the first factor G_{1} is the exceptional group G_{2}. The point is that, in this case, neither H_{3} = Sp(4) nor H_{2} (from the list, above) has a nontrivial homomorphism to the first factor G_{1}. Since H_{2} × H_{3} ⊂ H must act freely on G_{1}×G_{2}, it follows that H_{2}×H_{3} acts freely on the second factor G_{2}. This is impossible because the list of possible spaces G_{2}/H_{2} shows that H_{2} has rank 1 less than the second factor G_{2}, and so H_{2} × H_{3} = H_{2} × Sp(4) has rank 1 greater than the second factor G_{2}. The impossibility here follows from the fact that a finite-dimensional elliptic space X has the alternating sum of its rational homotopy groups χ_{π}(X) ≤ 0, by Halperin ([13], p. 434). So the first factor G_{1} is not the exceptional group G_{2}.

We continue to assume that H_{3} is Sp(4). We know that G_{1}/H_{1} is a homogeneous space Spin(12)/Spin(11) = SU(6)/SU(5) = Sp(6)/Sp(4) = S^{11} or Spin(7)/Spin(5) = UT(S^{6}), and G_{2}/H_{2} is a homogeneous space Spin(16)/Spin(15) = SU(8)/SU(7) = Sp(8)/Sp(6) = Spin(9)/Spin(7) = S^{15} or Spin(9)/Spin(7) = UT(S^{8}). For each possible G_{1}/H_{1} and G_{2}/H_{2}, we check immediately that either H_{1} must act trivially on G_{2} or H_{2} must act trivially on G_{1}. Here we use in particular that a simple factor of H must act trivially on a simple factor of G isomorphic to it, by Lemma 4.1. It follows that the biquotient (G_{1} ×G_{2})/(H_{1} ×H_{2}) is either a G_{1}/H_{1}-bundle over G_{2}/H_{2} or a G_{2}/H_{2}- bundle over G_{1}/H_{1}. Now G_{1}/H_{1} is either S^{11} or UT(S^{6}), so it has the 3-local homotopy type of S^{11}; and likewise G_{2}/H_{2} is either S^{15} or UT(S^{8}), so it has the 3-local homotopy type of S^{15}. Therefore the biquotient (G_{1} × G_{2})/(H_{1} × H_{2}) is 3-locally 11-connected. It follows that the natural map

M = (G_{1} × G_{2})/(H_{1} × H_{2} × Sp(4)) → BSp(4)

is 3-locally 11-connected. Up to this point, any odd prime number would serve in place of 3, but we now derive a contradiction by a 3local calculation. Since M is homotopy equivalent to HP^{4}#CaP^{2}, the map π_{8}M ⊗ H^{8}(M, Z) → Z is surjective. On the other hand, the map π_{8}BSp(4) × H^{8}(BSp(4), Z) → Z has image 6Z. Indeed, H^{8}(BSp(4), Z) is generated by c_{2}^{2} and c_{4}, and c_{2}^{2} is clearly zero for any Sp(4)-bundle over S^{8}, while c_{4} of such a bundle is a multiple of (4 − 1)! = 6 by Bott periodicity. Since the above map M → BSp(4) is 3-locally 11-connected, we have a contradiction.

That shows that H_{3} is not isomorphic to Sp(4). Therefore H_{3} is isomorphic to SU(2). In this case, we know that G_{2}/H_{2} is Spin(9)/Spin(8) = S^{8}. Also, G_{1}/H_{1} is either a homogeneous space Spin(12)/Spin(11) = SU(6)/SU(5) = Sp(6)/Sp(4) = S^{11} or Spin(7)/Spin(5) = UT(S^{6}), or else G_{1} is the exceptional group G_{2} and H_{1} is SU(2) acting nontrivially on G_{1}, perhaps on both sides.

Since M = G/H is homotopy equivalent to HP^{4}#CaP^{2}, it has the same Stiefel-Whitney classes as HP^{4}#CaP^{2}, by Wu ([19], Theorem 11.14). In particular, the sphere S^{8} in HP^{4}#CaP^{2} has self-intersection number 1, and so w_{8}M is not in the subgroup H^{4}(M, F_{2})^{2} of H^{8}(M, F_{2}). The biquotient N := (G_{1} × G_{2})/(H_{1} × H_{2}) is a principal SU(2)-bundle over M, so its stable tangent bundle is the pullback of that of M. It follows that w_{8}N ∈ H^{8}(N, F_{2}) is not zero.

Suppose that G_{1} is the exceptional group G_{2}. Then M is a biquotient of the form (G_{2}×Spin(9))/(SU(2)×Spin(8)×SU(2)). Here Spin(8) must act trivially on G_{2}, and so we can write

M = (G_{2} × S^{8})/SU(2)^{2}.

Any torus acting by isometries on an even-dimensional sphere has a fixed point, since every orthogonal representation of a torus is a sum of 2-dimensional representations and trivial representations. In particular, a maximal torus in SU(2)^{2} has a fixed point on S^{8}. Since SU(2)^{2} acts freely on G_{2} × S^{8}, this torus acts freely on G_{2}. Since every element of SU(2)^{2} is conjugate to an element of the torus, SU(2)^{2} acts freely on G_{2}. Thus M is a fiber bundle

S^{8} → M → G_{2}/SU(2)^{2}.

Since S^{8} has signature zero, it follows that M has signature zero, contradicting the fact that M is homotopy equivalent to HP^{4}#CaP^{2}. Thus the first factor G_{1} is not the exceptional group G_{2}.

So G_{1}/H_{1} is either Spin(12)/Spin(11) = SU(6)/SU(5) = Sp(6)/Sp(4) = S^{11} or Spin(7)/Spin(5) = UT(S^{6}), and we know that G_{2}/H_{2} is Spin(9)/Spin(8) = S^{8}. But since w_{8}N is not zero, N is not an S^{8}-bundle over S^{11} or over UT(S^{6}). So H_{2} = Spin(8) must act nontrivially on G_{1}. It follows that G_{1}/H_{1} must be Spin(12)/Spin(11) = S^{11}.

Thus M is a biquotient of the form

(Spin(12) × Spin(9))/(Spin(11) × Spin(8) × SU(2)).

Here Spin(11) automatically acts trivially on Spin(9). So the biquotient

N := (Spin(12) × Spin(9))/(Spin(11) × Spin(8))

is completely determined by the homomorphism Spin(8) → Spin(12), which we know is nontrivial. (Since Spin(11) has finite centralizer on Spin(12), the action of Spin(8) on Spin(12) must be on the other side of Spin(12) from Spin(11).) There are 3 conjugacy classes of nontrivial homomorphisms Spin(8) → Spin(12), each of the form W ⊕ R^{4}, where the 8-dimensional real representation W of Spin(8) is either the standard representation V or one of the two spin representations. Thus N is the S^{11}-bundle S(W ⊕ R^{4}) over Spin(9)/Spin(8) = S^{8}. If W is the standard representation V of Spin(8), then the associated rank-8 vector bundle over S^{8} has w_{8} = 0, and so the manifold N would have w_{8}N = 0, contradicting what we know. So W must be one of the two spin representations of Spin(8). If necessary, we can apply an order-2 automorphism to Spin(8), not changing the conjugacy class of the standard inclusion Spin(8) → Spin(9), to arrange that W = S^{−}.

Thus N is the S^{11}-bundle S(S^{−} ⊕ R^{4}) over S^{8}. The action of SU(2) on N is given by a homomorphism SU(2) → Spin(9) together with a homomorphism SU(2) → Spin(4), because Spin(4) is the identity component of the centralizer of Spin(8) in Spin(12). The homomorphism SU(2) → Spin(9) determines the action of SU(2) on S^{8} and on the Spin(9)-equivariant vector bundle S^{−} over S^{8}, while the homomorphism SU(2) → Spin(4) determines the action of SU(2) on R^{4}. Since SU(2) acts freely on N = S(S^{−} ⊕ R^{4}), it must act freely on S(S^{−}) = S^{15} and on S(R^{4}) = S^{3} × S^{8}.

As in the proof that CP^{8}#CaP^{2} is not homotopy equivalent to a biquotient, we compute that there is only one free SU(2)-action on S(S^{−} ⊕ R^{4}): SU(2) must act by SU(2) ≅ Spin(3) ⊂ Spin(9) on S^{8} and on the bundle S^{−} over S^{8}, and by the standard representation V_{R} on R^{4}. Thus we have described the manifold M uniquely up to diffeomorphism. But we showed in Section 2 that exactly this manifold is diffeomorphic to HP^{4}# − CaP^{2}. In particular, it is not homotopy equivalent to HP^{4}#CaP^{2}. This completes the proof that HP^{4}#CaP^{2} is not homotopy equivalent to a biquotient.

8. The Cheeger manifold CP^{4e}#HP^{2e} is not homotopy equivalent to a biquotient

Finally, we will show that CP^{4e}#HP^{2e}, e ≥ 1, is not homotopy equivalent to a biquotient M = G/K, completing the proof of Theorem 2.1.

The cohomology ring of M is a complete intersection ring. The degrees of its generators and relations determine the rational homotopy groups of M. The result is that G contributes degrees 3 and 4e to M and K contributes degrees 1 and 2, where the 1 means that the abelianization of K is isomorphic to S^{1}. Let H be the commutator subgroup of K, and let N be the biquotient G/H; then M = N/S^{1}. Here G contributes degrees 3 and 4e to N, while H contributes degree 2. From the degree 2, it follows that H has one more simple factor than G has. Also, by Theorem 4.8, each simple factor of G contributes at least one degree to N. So G has at most two simple factors.

Furthermore, since M and CP^{4e}#HP^{2e} are homotopy equivalent, they have the same Stiefel-Whitney classes, by Wu ([19], Theorem 11.14). In particular, w_{4}M is not in the subgroup H^{2}(M, F_{2})^{2} of H^{4}(M, F_{2}). Since N is an S^{1}-bundle over M, the stable tangent bundle of N is the pullback of that of M, and so w_{4}N is not zero.

Suppose first that G is simple. Since H has one more simple factor than G, H has two simple factors. We know that G contributes degrees 3 and 4e to N, and nothing else. In Theorem 4.8, cases (2), (3), (4) are incompatible with these degrees, and so case (1) must hold, that is, G contributes its maximal degree to N. Thus G has maximal degree 4e. The only simple groups which have 3 as a degree are the groups SU(n), n ≥ 3, and so G is isomorphic to SU(4e). Since G has exactly one degree 4e, the known degrees of N imply that all simple factors of H have maximal degree less than 4e.

Suppose that e ≥ 2. Then the second-largest degree, 4e − 1, of G = SU(4e) is killed by H. Let H_{1} be a simple factor of H which kills the degree 4e − 1 of G. We know that H_{1} has maximal degree less than 4e, and so it has maximal degree 4e−1, which implies that H_{1} is isomorphic to SU(4e − 1). Any nontrivial homomorphism SU(4e − 1) → SU(4e) has Dynkin index 1, and so H_{1} must act nontrivially on only one side of G; otherwise SU(4e) would contribute degree 2 to N. So G/H_{1} is isomorphic to the homogeneous space SU(4e)/SU(4e − 1) = S^{8e−1}. But then H_{1} kills the degree 3 of G, which should appear in M. Thus e ≥ 2 leads to a contradiction, for G simple.

This leaves the case e = 1, with G simple. As shown above, G is isomorphic to SU(4). Since G has degrees 2, 3, 4, the known degrees of N imply that H has degrees 2, 2. So H is isomorphic to SU(2)^{2}. Thus N is a biquotient SU(4)/SU(2)^{2}. Indeed, there is at least one such biquotient, the homogeneous space UT(S^{5}) = Spin(6)/Spin(4), which admits a free S^{1}-action. In that case, the quotient is the 4-dimensional complex quadric Q_{C}^{4}, which has the rational homotopy type of CP^{4}#HP^{2}. But no biquotient N = SU(4)/SU(2)^{2} can have the homotopy type of an S^{1}-bundle over CP^{4}#HP^{2}, by the following argument. By Singhof’s description of the tangent bundle of a biquotient [25], as used in the previous section, we have

w_{4}N = w_{4}(su(4)) − w_{4}(su(2)_{1}) − w_{4}(su(2)_{2}).

We then make the convenient calculation that

w_{4}(su(n)) = 0 ∈ H^{4}(BSU(n), F_{2})

for all even n. So w_{4}N = 0, which contradicts what we know about N. Thus we have a contradiction from the assumption that G is simple.

Thus G has two simple factors. We return to the general case, e ≥ 1. Since H has one more simple factor than G, H has three simple factors. Each simple factor of G contributes exactly one degree to N. We can write G = G_{1} × G_{2} such that G_{1} contributes degree 3 to N and G_{2} contributes degree 4e, and nothing more. By Theorems 4.8 and 5.3 applied to G_{1}, there is a simple factor H_{1} of H such that either G_{1}/H_{1} is isomorphic to the homogeneous space SU(4)/Sp(4) = S^{5} (which is the same as Spin(6)/Spin(5)) or (G_{1}, H_{1}) is (SU(3), SU(2)) for some nontrivial action of SU(2) on SU(3).

If (G_{1}, H_{1}) is (SU(3), SU(2)) and SU(2) acts nontrivially on both sides of SU(3), then the centralizers of both homomorphisms SU(2) → SU(3) are finite or finite by S^{1}, so no other simple factor of H can act on G_{1}. Moreover, the two nontrivial homomorphisms SU(2) → SU(3) have Dynkin indices 1 and 4, by Lemma 5.2. So the map π_{3}H_{1} → π_{3}G_{1}, Z → Z, is either zero or multiplication by 1−4 = −3 or 4−1 = 3. In any case, it is not surjective, which contradicts the fact that π_{3}H → π_{3}G_{1} must be surjective, since π_{3}M = 0. So any SU(2) factor of H acts nontrivially on at most on one side of SU(3). Moreover, SU(2) factors of H are the only ones that can act on G_{1} = SU(3) (SU(3) factors of H are excluded by Lemma 4.1). At least one SU(2) factor must act with Dynkin index 1 rather than 4, again using that π_{3}H → π_{3}G_{1} is surjective. Thus, if G_{1} is SU(3), then we can choose the simple factor H_{1} isomorphic to SU(2) such that G_{1}/H_{1} is isomorphic to the homogeneous space SU(3)/SU(2) = S^{5}.

Thus, G_{1}/H_{1} is either Spin(6)/Spin(5) = S^{5} or SU(3)/SU(2) = S^{5}. In both cases, G_{1}/H_{1} adds degree 3 to N and nothing else. Next, we can apply Theorems 4.8 and 5.3 to the second factor G_{2}, giving a simple factor H_{2} with certain properties. From the known degrees of N = (G_{1} ×G_{2})/(H_{1} ×H_{2} ×H_{3}), since G_{1}/H_{1} adds degree 3 to N and nothing else, G_{2}/H_{2} cannot have any degrees with multiplicity < 0 (that is, any degrees which occur in H_{2} more than in G_{2}). Given this, Theorems 4.8 and 5.3 imply that either e = 1 and G_{2} is isomorphic to Sp(4), or there is a simple factor H_{2} of H such that G_{2}/H_{2} is isomorphic to a homogeneous space Spin(8e)/Spin(8e−1) = SU(4e)/SU(4e−1) = Sp(4e)/Sp(4e−2) = S^{8e−1}, Spin(7)/G_{2} = S^{7} with e = 1, Spin(9)/Spin(7) = S^{15} with e = 2, Spin(4e + 1)/Spin(4e − 1) = UT(S^{4e}) with e ≥ 2, Spin(8)/G_{2} = S^{7} × S^{7} with e = 1, Spin(9)/G_{2} with e = 2, Spin(4e+2)/Spin(4e−1) with e ≥ 2, Spin(10)/Spin(7) via the spin representation with e = 2, or Spin(10)/G_{2} with e = 2. In these homogeneous spaces, it is clear that the simple factor H_{2} of H is different from H_{1}, which is SU(2) or Spin(5) = Sp(4).

When G_{2}/H_{2} is one of the homogeneous spaces diffeomorphic to S^{8e−1} or UT(S^{4e}), then G_{2}/H_{2} adds only one degree 4e to N and nothing else. In those cases, the third factor H_{3} of H has degree 2 only, so H_{3} is isomorphic to SU(2). When G_{2} is isomorphic to Sp(4), the product of the two simple factors of H besides H_{1}, H_{2} ×H_{3}, has degrees 2, 2, and so H_{2} and H_{3} are both isomorphic to SU(2). When G_{2}/H_{2} is Spin(8)/G_{2} = S^{7} × S^{7} with e = 1 or Spin(9)/G_{2} with e = 2, then H_{3} has degrees 2, 4, and so H_{3} is isomorphic to Sp(4). Finally, the cases where G_{2}/H_{2} is Spin(4e + 2)/Spin(4e − 1) with e ≥ 2, Spin(10)/Spin(7) with e = 2, or Spin(10)/G_{2} with e = 2 cannot occur. In these cases, part (4) of Theorem 5.3 implies that H must have a factor SU(2e + 1), which must be the third factor H_{3}. But then H has one more degree 3 than G, contradicting the known degrees of N = G/H.

The cases where H_{3} is Sp(4) are easier to analyze, so we consider them first. Either G_{2}/H_{2} is Spin(8)/G_{2} = S^{7} × S^{7} and e = 1, or G_{2}/H_{2} is Spin(9)/G_{2} and e = 2. We know that G_{1}/H_{1} is either SU(3)/SU(2) = S^{5} or Spin(6)/Spin(5) = S^{5}. By the known low-dimensional representations of H_{3} = Sp(4), the action of H_{3} on G_{1}/H_{1} = S^{5} must have a fixed point. So there is a subgroup of H_{1} × H_{3} which projects isomorphically to H_{3} = Sp(4) and which fixes a point in G_{1}. Also, the exceptional group G_{2} must act trivially on G_{1} (which is Spin(6) or SU(3)), so we have a subgroup of H isomorphic to Sp(4) × G_{2} which fixes a point in G_{1}. Since H acts freely on G, we have a free action of Sp(4) × G_{2} on the second factor Spin(8) or Spin(9). We compute, however, that there is no such free action.

So we must have H_{3} = SU(2). We know that G_{1}/H_{1} is either SU(3)/SU(2) = S^{5} or Spin(6)/Spin(5) = S^{5}. Also, either e = 1, G_{2} is isomorphic to Sp(4) and H_{2} is isomorphic to SU(2), or G_{2}/H_{2} is a homogeneous space Spin(8e)/Spin(8e − 1) = SU(4e)/SU(4e − 1) = Sp(4e)/Sp(4e − 2)=S^{8e−1}, Spin(7)/G_{2} =S^{7} with e=1, Spin(9)/Spin(7) = S^{15} with e = 2, or Spin(4e + 1)/Spin(4e − 1) = UT(S^{4e}) with e ≥ 2.

We begin with the case where G_{1}/H_{1} is SU(3)/SU(2) = S^{5} and G_{2}/H_{2} is Spin(8e)/Spin(8e − 1) = S^{8e−1}. This turns out to be the main step of the whole proof; most other cases will reduce to this one. The group H_{2} = Spin(8e − 1) must act trivially on G_{1} = SU(3), and so (G_{1}×G_{2})/(H_{1}×H_{2}) is an S^{8e−1}-bundle over S^{5}. The manifold M is the quotient of this bundle by a free action of a group D := K/(H_{1} × H_{2}) which is isogenous to S^{1} × SU(2). Since π_{2}M is isomorphic to Z, π_{1}D is isomorphic to Z. So D is isomorphic to S^{1} × SU(2) or to U(2).

In dimensions less than 8e−1, M clearly has the homotopy type of a homotopy quotient S^{5}//D, or equivalently an S^{5}-bundle over the classifying space BD. Here D acts on S^{5} through the group U(3), coming from the group G_{1} = SU(3) together with the centralizer S^{1} of H_{1} = SU(2) in G_{1}. From the cohomology ring of M, the Euler class of the homomorphism D → U(3) must be the product of a generator of H^{2}(BD, Z) with an element of H^{4}(BD, Z) that generates H^{4}/(H^{2})^{2}. Let L be the standard 1-dimensional complex representation of S^{1}, V be the standard 2dimensional representation of SU(2), and E the standard 2-dimensional representation of U(2). By inspecting the low-dimensional representations of D, it follows that the homomorphism D → U(3) is isomorphic to L^{±1} ⊕ L^{a} ⊗ V if D is S^{1} × SU(2), or to (det E)^{±1} ⊕ (det E)^{a} ⊗ E if D is U(2), for some sign and some integer a. In particular, the subgroup SU(2) = H_{3} of D acts on G_{1}/H_{1} = SU(3)/SU(2) = S^{5} by the natural inclusion SU(2) → SU(3), on the other side of G_{1} from H_{1}.

It follows that the diagonal subgroup C in H_{1} ×H_{3} = SU(2)×SU(2) has a fixed point in SU(3). So C must act freely on S^{8e−1}. It follows that C acts on S^{8e−1} by the real representation (V_{R})^{2e}, where V is the standard 2-dimensional complex representation of C = SU(2). In particular, the associated complex representation is a direct sum of copies of V. By the Clebsch-Gordan formula, an irreducible complex representation of SU(2) × SU(2) whose restriction to the diagonal subgroup is a sum of copies of V must be isomorphic to V_{1} ⊗ C or C ⊗ V_{2}, the standard representations of the two factors. Therefore the action of H_{1} × H_{3} = SU(2) × SU(2) on S^{8e−1} must be given by the real representation associated to the complex representation (V_{1})^{⊕j} ⊕ (V_{3})^{⊕2e−j} for some 0 ≤ j ≤ 2e.

The manifold M can be written

M = (SU(3) × S^{8e−1})/((SU(2)^{2} × R)/Z).

Here the subgroup Z of SU(2)^{2} × R is generated by an element of the form (±1, ±1, 1), which we write as (e(a_{0}), e(b_{0}), 1) for a_{0}, b_{0} ∈ {0, 1/2}, where e(t) := e^{2πit}. The group (SU(2)^{2} × R)/Z acts on SU(3) × S^{8e−1} via homomorphisms to SU(3)^{2}/Z(SU(3)) and to SO(8e).

We can assume that the first factor SU(2) of (SU(2)^{2} × R)/Z acts on SU(3) by the standard inclusion on the left, while the second factor SU(2) acts on SU(3) by the standard inclusion on the right. So the two homomorphisms from R to SU(3) both map into the centralizer of SU(2) in SU(3). That is, they map t ∈ R to the diagonal matrices

(e(at), e(at), e(−2at)), (e(bt), e(bt), e(−2bt))

for some a, b. Since the generator of the above subgroup Z in SU(2)^{2}×R must map into Z(SU(3)) ≅ Z/3 ⊂ SU(3)^{2}, a and b must satisfy:

a + a_{0} ∈ \frac{1}{3}Z
b + b_{0} ∈ \frac{1}{3}Z
a + a_{0} ≡ b + b_{0} (mod Z).

The homomorphism SU(2)^{2} → SO(8e) is by the real representation (V_{1})^{⊕j} ⊕ (V_{2})^{⊕2e−j} for some 0 ≤ j ≤ 2e. The centralizer of this homomorphism has identity component Sp(2j)×Sp(4e−2j), whose maximal torus is conjugate to the center (S^{1})^{2e} of U(2)^{2e}. So we can assume that R maps into this center. Thus the homomorphism from (SU(2)^{2}×R)/Z to SO(8e) is the direct sum of 2e homomorphisms to U(2), of the form (A, B, t) ↦ e(c_{i}t)A for 1 ≤ i ≤ j and (A, B, t) ↦ e(d_{i}t)B for j + 1 ≤ i ≤ 2e. Since this homomorphism is trivial on the subgroup Z, the numbers c_{i} and d_{i} must satisfy c_{i} + a_{0} ∈ Z and d_{i} + b_{0} ∈ Z.

The action of (SU(2)^{2} ×R)/Z on SU(3)×S^{8e−1} is free. We compute that this means that, for all 1 ≤ i ≤ j,

a + 2b − c_{i} = ±1
a + 2b + c_{i} = ±1
−2a + 2b = ±1,

and for all j + 1 ≤ i ≤ 2e,

2a + b − d_{i} = ±1
2a + b + d_{i} = ±1
−2a + 2b = ±1.

If j ≥ 1, then the first two equations imply that either a + 2b = 0 and c_{i} = ±1, or a + 2b = ±1 and c_{i} = 0, for 1 ≤ i ≤ j. In particular, c_{i} is an integer, which implies (since c_{i} + a_{0} ∈ Z) that a_{0} is 0, not 1/2. Since a + a_{0} ∈ (1/3)Z, it follows in particular that a is 2-integral; since −2a + 2b = ±1, b is not 2-integral. Therefore b_{0} is 1/2, not 0. But if j ≤ 2e−1, then we would get the opposite conclusion (that a_{0} = 1/2 and b_{0} = 1) from the second three equations above. So we must have either j = 0 or j = 2e. After switching the two SU(2) factors if necessary, we can assume that j = 2e. That is, SU(2)^{2} acts on S^{8e−1} by the real representation (V_{1})^{2e}. Also, we have a_{0} = 0 and b = 1/2, which means that the subgroup Z of SU(2)^{2} × R is generated by (1, −1, 1 ∈ R).

Since the second factor SU(2) acts only on SU(3), we can rewrite M as

M = (S^{5} × S^{8e−1})/(SU(2) × S^{1}).

Here we have used that the quotient of (SU(2)^{2} × R)/Z by the second copy of SU(2) is isomorphic to SU(2) × S^{1}. The action of SU(2) × S^{1} on S^{8e−1} is given by the complex representation ⊕_{i=1}^{2e}V ⊗ L^{c_{i}}. Also, we compute that the action of SU(2) × S^{1} on S^{5} is by the complex representation (V ⊗ L^{a+2b}) ⊕ L^{−2a+2b}. As computed above, −2a + 2b = ±1. And either a + 2b = ±1 and c_{i} = 0 for all i, or a + 2b = 0 and c_{i} = ±1 for all i.

In the first case, where c_{i} = 0, we can conjugate the action of SU(2)× S^{1} on S^{5} in the orthogonal group O(6) to make a + 2b and −2a + 2b equal to 1, rather than −1. Thus M is the manifold

(S((V ⊗ L) ⊕ L) × S(V^{⊕2e}))/(SU(2) × S^{1}).

So M is a CP^{2}-bundle over HP^{2e−1}, and so M has signature zero, contradicting that M is homotopy equivalent to CP^{4e}#HP^{2e}. In fact, this CP^{2}-bundle over HP^{2e−1} is the one diffeomorphic to CP^{4e}# − HP^{2e}, as mentioned in Section 2. In the second case, where c_{i} = ±1, we can conjugate the homomorphisms from SU(2)×S^{1} to the orthogonal groups O(6) and O(8e) to make −2a + 2b = 1 and c_{i} = 1 for all i. Thus, M is the manifold

(S(V ⊕ L) × S((V ⊗ L)^{⊕2e}))/(SU(2) × S^{1}).

Again, we showed in Section 2 that this manifold is diffeomorphic to CP^{4e}# − HP^{2e}. Thus it is not homotopy equivalent to CP^{4e}#HP^{2e}. This completes the proof that we cannot have G_{1}/H_{1} equal to SU(3)/SU(2) = S^{5} and G_{2}/H_{2} equal to Spin(8e)/Spin(8e − 1) = S^{8e−1}.

We next consider the case where G_{1}/H_{1} is Spin(6)/Spin(5) = S^{5} and G_{2}/H_{2} is Spin(8e)/Spin(8e − 1) = S^{8e−1}. A first observation is that Spin(5) has finite centralizer in Spin(6), and so all factors of K except Spin(5) act on the other side of Spin(6) from Spin(5). Likewise, all factors of K except Spin(8e−1) act on the other side of Spin(8e) from Spin(8e − 1). Furthermore, Spin(8e − 1) must act trivially on Spin(6), and so (G_{1} × G_{2})/(H_{1} × H_{2}) is an S^{8e−1}-bundle over S^{5}, which we can write as (Spin(6) × S^{8e−1})/Spin(5).

The homomorphism K → (G × G)/Z(G) which defines the action of K on G gives a homomorphism from D := K/(H_{1} ×H_{2}) to SO(6). Here D is isogenous to SU(2) × S^{1} and has fundamental group isomorphic to Z, so D is isomorphic to SU(2) × S^{1} or to U(2). In dimensions less than 8e−1, M has the homotopy type of the homotopy quotient S^{5}//D defined by the homomorphism D → SO(6), or equivalently of an S^{5}- bundle over the classifying space BD. From the cohomology ring of M, the Euler class of the homomorphism D → SO(6) must be the product of some generator of H^{2}(BD, Z) with some element of H^{4}(BD, Z) which generates H^{4}/(H^{2})^{2}. By inspecting the low-dimensional real representations of D, it follows that the homomorphism D → SO(6) is the real representation associated to a 3-dimensional complex representation of D. For D = S^{1}×SU(2), write L for the standard 1-dimensional complex representation of S^{1} and V for the standard 2-dimensional complex representation of SU(2). In order to have an Euler class of the form above, the homomorphism D → SO(6) must come from the complex representation L^{±1} + L^{b} ⊗ V for some sign and some integer b. The Euler class of this representation is ±xy, where we let x = c_{1}L and y = c_{2}V + b^{2}x^{2}. Likewise, for D = U(2), let E be the standard 2-dimensional complex representation of D. In order to have an Euler class of the form above, the homomorphism D → SO(6) must come from the complex representation (det E)^{±1} + (det E)^{d} ⊗ E for some sign and some integer d. The Euler class in H^{6}BU(2) of this representation is ±uv, where we let u = c_{1}E and v = c_{2}E + (d^{2} + d)u^{2}.

Because the homomorphism D → SO(6) factors through U(3), we can replace G_{1}/H_{1} = Spin(6)/Spin(5) = S^{5} by G_{1}/H_{1} = Ũ(3)/Ũ(2), in our description of M as a biquotient. Here Ũ(n) denotes the inverse image of U(n) ⊂ SO(2n) in Spin(2n). We can then apply the proof of Lemma 3.1 to replace Ũ(3) by SU(3). Thus we have reduced to the case where G_{1}/H_{1} is SU(3)/SU(2) = S^{5} and G_{2}/H_{2} is Spin(8e)/Spin(8e − 1) = S^{8e−1}. But we have shown that the latter case cannot occur. This completes the proof that G_{2}/H_{2} cannot be Spin(8e)/Spin(8e − 1) = S^{8e−1}, either when G_{1}/H_{1} is Spin(6)/Spin(5) or when it is SU(3)/SU(2).

The situation now is as follows. First, we know that G_{1}/H_{1} is either SU(3)/SU(2) = S^{5} or Spin(6)/Spin(5) = S^{5}. Also, either e = 1, G_{2} is isomorphic to Sp(4) and H_{2} is isomorphic to SU(2), or G_{2}/H_{2} is a homogeneous space Spin(8e)/Spin(8e − 1) = SU(4e)/SU(4e − 1) = Sp(4e)/Sp(4e − 2)=S^{8e−1}, Spin(7)/G_{2} =S^{7} with e=1, Spin(9)/Spin(7) = S^{15} with e = 2, or Spin(4e + 1)/Spin(4e − 1) = UT(S^{4e}) with e ≥ 2. We have shown that G_{2}/H_{2} cannot be Spin(8e)/Spin(8e − 1) = S^{8e−1}, either when G_{1}/H_{1} is Spin(6)/Spin(5) or when it is SU(3)/SU(2). Most other cases reduce to this one. Namely, suppose that G_{2}/H_{2} is diffeomorphic to a sphere S^{8e−1} (noting that in all these cases G_{2} × Z_{G_{2}}(H_{2}) acts by isometries in the usual metric on the sphere), and that H_{2} acts trivially on G_{1}. Then we can simply replace G_{2}/H_{2} by Spin(8e)/Spin(8e − 1) = S^{8e−1} without changing the biquotient M. Since G_{1} is small, namely Spin(6) or SU(3), the hypothesis that H_{2} acts trivially on G_{1} is automatic for most of the pairs G_{2}/H_{2}. Namely this holds when there is no nontrivial homomorphism H_{2} → G_{1}, or also when H_{2} is isomorphic to G_{1} by Lemma 4.1.

The cases not covered by this argument are: G_{1}/H_{1} is SU(3)/SU(2) = S^{5} or Spin(6)/Spin(5) = S^{5}, G_{2} is isomorphic to Sp(4), and H_{2} is isomorphic to SU(2), where e = 1; G_{1}/H_{1} is Spin(6)/Spin(5) and G_{2}/H_{2} is SU(4)/SU(3) = S^{15} with H_{2} acting nontrivially on G_{1}, where e = 2; or G_{1}/H_{1} is SU(3)/SU(2) or Spin(6)/Spin(5) and G_{2}/H_{2} is Spin(4e + 1)/Spin(4e − 1) = UT(S^{4e}) with e ≥ 2.

The last case, where G_{2}/H_{2} is UT(S^{4e}) with e ≥ 2, is easy to exclude. Let N = (G_{1} × G_{2})/(H_{1} × H_{2} × H_{3}), so that M = N/S^{1}. Because the groups involved in N are simply connected, N is 2-connected, and so N is the S^{1}-bundle over M corresponding to a generator of H^{2}(M, Z). Since M has the integral cohomology ring of CP^{4e}#HP^{2e}, the spectral sequence of this S^{1}-bundle shows that N has the integral cohomology ring of S^{5} ×HP^{2e−1}. Next, let Y be the manifold (G_{1} ×G_{2})/(H_{1} ×H_{2}), which is an SU(2)-bundle over N because H_{3} = SU(2). Because H_{2} = Spin(4e − 1) and e ≥ 2, H_{2} acts trivially on G_{1} (which is Spin(6) or SU(3)), and so Y is a UT(S^{4e})-bundle over S^{5}. In particular, Y is 4connected. Also, the spectral sequence computing the cohomology of Y collapses for degree reasons, and so Y has 2-torsion in its cohomology because UT(S^{4e}) does. But Y is also an S^{3}-bundle over N, and because Y is 4-connected, the Euler class of this bundle must be a generator of H^{4}(N, Z). So the spectral sequence of this S^{3}-bundle shows that Y has the integral cohomology ring of S^{5} × S^{8e−1}. This contradicts the fact that Y has 2-torsion. So this case, G_{2}/H_{2} = UT(S^{4e}) with e ≥ 2, does not occur.

Next, we consider the case where G_{1}/H_{1} is Spin(6)/Spin(5) = S^{5} and G_{2}/H_{2} is SU(4)/SU(3) = S^{15}, with SU(3) acting nontrivially on Spin(6). Here e = 2. The point is that any nontrivial action of SU(3) on S^{5} is isomorphic to the standard action, and hence is transitive. Thus H acts transitively on the factor G_{1} of G, contrary to Convention 3.4.

This completes the proof that CP^{4e}#HP^{2e} is not homotopy equivalent to a biquotient for all e ≥ 2. Ironically, the hardest case of all is the case e = 1, that is, the proof that CP^{4}#HP^{2} is not homotopy equivalent to a biquotient.

For e = 1, it remains to consider the case where G_{1}/H_{1} is either SU(3)/SU(2) = S^{5} or Spin(6)/Spin(5) = S^{5}, G_{2} is isomorphic to Sp(4), and H_{2} is isomorphic to SU(2). Thus the 9-manifold N is a biquotient of the form

(SU(3) × Sp(4))/SU(2)^{3}

or

(Spin(6) × Sp(4))/(Spin(5) × SU(2)^{2}),

where the first factor H_{1} acts by the standard inclusion on one side of G_{1} and trivially on the other side of G_{1}. Also, if H_{2} or H_{3} (each isomorphic to SU(2)) acts trivially on G_{1}, and acts trivially on one side of Sp(4) and by the standard inclusion V ⊕ C^{2} on the other, then we can replace the quotient Sp(4)/SU(2) = S^{7} by Spin(8)/Spin(7) = S^{7} and thus reduce to an earlier case. So we can assume that neither H_{2} nor H_{3} has these properties.

We checked earlier that the Stiefel-Whitney class w_{4}N is not zero. It is convenient to observe now that the Pontrjagin class p_{1}(HP^{2}) is 2z, where z is a generator of H^{4}(HP^{2}, Z) ≅ Z. Furthermore, Wu showed that the first Pontrjagin class of a closed manifold is a homotopy invariant modulo 12 [28]. Since M is homotopy equivalent to CP^{4}#HP^{2}, p_{1}M in H^{4}M/(H^{2}M)^{2} ≅ Z is 2 times the class of some generator, modulo 12. Therefore, the S^{1}-bundle N over M has p_{1}N equal to 2 times the class of some generator of H^{4}N ≅ Z, modulo 12.

Lemma 8.1. If H_{1} acts trivially on G_{2} = Sp(4), then H_{2} × H_{3} = SU(2)^{2} does not act freely on G_{2} = Sp(4).

Proof. Suppose that H_{1} acts trivially on Sp(4) and that H_{2} × H_{3} acts freely on Sp(4). Since H_{1} acts trivially on Sp(4), we can enlarge G_{1} and H_{1} if necessary to make G_{1}/H_{1} equal to Spin(6)/Spin(5), rather than SU(3)/SU(2). The quotient Sp(4)/(H_{2} × H_{3}) is diffeomorphic to S^{4}, by Lemma 6.2. Also, N is the S^{5}-bundle over S^{4} associated to some homomorphism from H_{2} × H_{3} to Spin(6). By our knowledge of p_{1}N, p_{1} of this S^{5}-bundle in H^{4}(S^{4}, Z) ≅ Z must be 2 times some generator, modulo 12.

By Lemma 6.2, after switching H_{2} and H_{3} and switching the two sides of Sp(4) if necessary, H_{2} ×H_{3} acts on Sp(4) by either (V_{2} ⊕V_{3}, C^{4}) or (V_{2} ⊕ C^{2}, (V_{3})^{⊕2}). First suppose that H_{2} × H_{3} acts on Sp(4) by (V_{2} ⊕ V_{3}, C^{4}). The conjugacy classes of homomorphisms from H_{2} × H_{3} to Spin(6) have complexifications: C^{6}, (S^{2}V_{2})^{⊕2}, S^{2}V_{2}+S^{2}V_{3}, (S^{2}V_{3})^{⊕2}, (V_{2})^{⊕2}⊕C^{2}, (V_{3})^{⊕2}⊕C^{2}, V_{2}⊗V_{3}⊕C^{2}, S^{4}V_{2}⊕C, and S^{4}V_{3}⊕C. The given homomorphism from H_{2}×H_{3} to Spin(6) must have first Pontrjagin class (that is, −c_{2} of the complexification) equal to 2 times some generator of H^{4}(S^{4}, Z) modulo 12, where both c_{2}V_{2} and c_{2}V_{3} represent the same generator of H^{4}(S^{4}, Z). This only occurs when the complexification is (V_{2})^{⊕2} ⊕ C^{2} or (V_{3})^{⊕2} ⊕ C^{2}. But then one of H_{2} or H_{3} acts trivially on Spin(6) and by (V ⊕ C^{2}, C^{4}) on Sp(4), contrary to what we arranged before the lemma.

It remains to consider the case where H_{2} × H_{3} acts on Sp(4) by (V_{2}⊕C^{2}, (V_{3})^{⊕2}). In this case, c_{2}V_{3} represents a generator of H^{4}(S^{4}, Z), while c_{2}V_{2} is 2 times that generator. Going through the list of possible homomorphisms from H_{2} × H_{3} to Spin(6) again, we find that the only one whose Pontrjagin class in H^{4}(S^{4}, Z) is 2 times a generator, modulo 12, is the one with complexification (V_{3})^{⊕2} ⊕ C^{2}. But then H_{2} acts trivially on Spin(6) and by (V ⊕ C^{2}, C^{4}) on Sp(4), contrary to what we arranged before the lemma. q.e.d.

We can now show that the case where G_{1}/H_{1} is Spin(6)/Spin(5) does not occur. In this case, we know that H_{1} acts trivially on Sp(4) (which is isomorphic to Spin(5)), by Lemma 4.1. By Lemma 8.1, H_{2}×H_{3} does not act freely on Sp(4). On the other hand, the action of H_{2}×H_{3} on Spin(6) is given by one of the homomorphisms listed in the proof of Lemma 8.1. In all these cases, we check immediately that the maximal torus (S^{1})^{2} in H_{2}×H_{3} = SU(2)^{2} has a fixed point in Spin(6)/Spin(5) = S^{5}. Therefore, there is a subgroup of H = H_{1} × H_{2} × H_{3} which projects isomorphically to (S^{1})^{2} in H_{2} × H_{3} and which has a fixed point on Spin(6). Since H acts freely on G, this subgroup acts freely on Sp(4). Since H_{1} acts trivially on Sp(4), this means that the maximal torus (S^{1})^{2} in H_{2} × H_{3} acts freely on Sp(4). Since every element of H_{2} × H_{3} is conjugate to an element of the maximal torus, it follows that H_{2} × H_{3} acts freely on Sp(4), contradicting what we have shown. Thus the case where G_{1}/H_{1} is Spin(6)/Spin(5) does not occur.

It remains to consider the case where G_{1}/H_{1} is SU(3)/SU(2) = S^{5}. That is, the 9-manifold N is a biquotient

(SU(3) × Sp(4))/SU(2)^{3},

where we know that H_{1} acts on SU(3) by the standard inclusion on one side, and we arranged earlier that neither H_{2} nor H_{3} acts trivially on one side of Sp(4) and by the standard inclusion on the other.

Suppose first that H_{2} and H_{3} act trivially on G_{1}. Then H_{2} × H_{3} ≅ SU(2)^{2} must act freely on Sp(4). By Lemma 6.2, up to switching H_{2} and H_{3} and switching the two sides of Sp(4), H_{2} × H_{3} acts on Sp(4) by the homomorphisms (V_{2} ⊕ C^{2}, V_{3} ⊕ V_{3}) or (V_{2} ⊕ V_{3}, C^{4}). Therefore, one of the factors H_{2} or H_{3} acts on Sp(4) trivially on one side and by the standard inclusion on the other, as well as acting trivially on SU(3). This contradicts what we arranged earlier.

So one of H_{2} or H_{3} acts nontrivially on G_{1}. Switching H_{2} and H_{3} if necessary, we can assume that H_{2} acts nontrivially on G_{1}. Since the centralizer of H_{1} = SU(2) in G_{1} = SU(3) is only finite by S^{1}, H_{2} acts only on the other side of G_{1} from H_{1}. It must act on G_{1} = SU(3) by the homomorphism V_{2} ⊕ C or S^{2}V_{2}. Thus the centralizer of H_{2} in G_{1} is at most finite by S^{1}. Since the centralizer of H_{1} on the other side of G_{1} is finite by S^{1}, H_{3} = SU(2) must act trivially on G_{1}. By what we arranged earlier, it follows that H_{3} does not act on Sp(4) trivially on one side and by the standard inclusion on the other.

Suppose first that H_{2} acts on SU(3) by V_{2} ⊕ C. Then H_{2} has a fixed point on S^{5}. More precisely, we see that the diagonal subgroup Δ_{12} ≅ SU(2) in H_{1} × H_{2} has a fixed point in SU(3). Since H_{3} acts trivially on G_{1} = SU(3), it follows that Δ_{12} × H_{3} ≅ SU(2)^{2} acts freely on G_{2} = Sp(4). By the classification of free actions of SU(2)^{2} on Sp(4) in Lemma 6.2, together with the fact that H_{3} does not act trivially on one side of Sp(4) and by the standard inclusion on the other, Δ_{12} ×H_{3} must act on Sp(4) by the homomorphism V_{12} ⊕C^{2} on one side and V_{3} ⊕V_{3} on the other. Here V_{12} denotes the standard representation of Δ_{12} = SU(2). By the Clebsch-Gordan formula, it follows that H_{1} × H_{2} × H_{3} = SU(2)^{3} acts on Sp(4) by V_{1} ⊕ C^{2} or V_{2} ⊕ C^{2} on one side and by V_{3} ⊕ V_{3} on the other. Switching H_{1} and H_{2} if necessary (which we can do, since they act the same way on G_{1} = SU(3)), we can assume that H_{1} × H_{2} × H_{3} = SU(2)^{3} acts on Sp(4) by V_{2} ⊕ C^{2} on one side and by V_{3}^{⊕2} on the other. Thus, H_{1} acts trivially on Sp(4) and H_{2} ×H_{3} acts freely on Sp(4), which contradicts Lemma 8.1.

Therefore H_{2} = SU(2) must act on G_{1} = SU(3) by the homomorphism S^{2}V_{2}. By Singhof’s description of the tangent bundle of a biquotient [25], w_{4}N is given by

w_{4}N = w_{4}su(3) + w_{4}sp(4) − w_{4}su(2)_{1} − w_{4}su(2)_{2} − w_{4}su(2)_{3}
= w_{4}su(3) + w_{4}sp(4),

using that w_{4}su(2) = 0 in H^{4}(BSU(2), F_{2}). Furthermore, we know that H_{2} acts on one side of SU(3) by S^{2}V_{2}, and no other factor of H acts on that side of SU(3). Since c_{2}(S^{2}V_{2}) = 4c_{2}V_{2}, the generator c_{2} of H^{4}(BSU(3), F_{2}) pulls back to 4c_{2}V_{2} = 0 in H^{4}(N, F_{2}). Therefore

w_{4}N = w_{4}sp(4).

We know that H_{3} acts trivially on SU(3), so it must act freely on Sp(4). Furthermore, we know that it does not act trivially on one side of Sp(4) and by the standard inclusion V_{3} ⊕ C^{2} on the other. By Lemma 6.2, H_{3} must act on at least one side of Sp(4) by the homomorphism V_{3}^{⊕2} or S^{3}V_{3}. These two homomorphisms have centralizers in Sp(4) which are finite by S^{1} or finite, so no other simple factor of H acts on the same side of Sp(4). Since c_{2}(V_{3}^{⊕2}) = 2c_{2}V_{3} and c_{2}(S^{3}V_{3}) = 10c_{2}V_{3}, the generator c_{2} of H^{4}(BSp(4), F_{2}) pulls back to 2c_{2}V_{3} or 10c_{2}V_{3} in H^{4}(N, F_{2}), thus to zero. Therefore

w_{4}N = 0,

contradicting what we know about N. This completes the proof that CP^{4}#HP^{2} is not homotopy equivalent to a biquotient. Theorem 2.1 is proved. q.e.d.

References

[1] J.F. Adams & M. Atiyah, K-theory and the Hopf invariant, Quart. J. Math. 17 (1966) 31–38, MR 33 #6618, Zbl 0136.43903.

[2] R. Aguilar, Symplectic reduction and the complex homogeneous Monge-Ampère equation, Ann. Global Anal. Geom. 19 (2001) 327–353, MR 2002g:53144, Zbl 1001.53058.

[3] M. Berger, Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive, Ann. Scuola Norm. Sup. Pisa 15 (1961) 179–246, MR 24 #A2919, Zbl 0101.14201.

[4] A. Borel, Le plan projectif des octaves et les sphères comme espaces homogènes, C.R. Acad. Sci. 230 (1950) 1378–1380, MR 11,640c, Zbl 0041.52203.

[5] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Masson, 1981, MR 39 #1590, Zbl 0186.33001.

[6] J. Cheeger, Some examples of manifolds of nonnegative curvature, J. Differential Geom. 8 (1973) 623–628, MR 49 #6085, Zbl 0281.53040.

[7] A. Dold, Erzeugende der Thomschen Algebra ℜ, Math. Zeit. 65 (1956) 25–35, MR 18,60c, Zbl 0071.17601.

[8] E. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik 30 (1952) 349–462, MR 13,904c; Amer. Math. Soc. Translations (2) 6 (1957) 111– 244, Zbl 0077.03404.

[9] J. Eschenburg, New examples of manifolds with strictly positive curvature, Invent. Math. 66 (1982) 469–480, MR 83i:53061, Zbl 0484.53031.

[10] J. Eschenburg, Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekrümmten Orbiträumen, Schriften der Math. Universität Münster 32, 1984, MR 86a:53045, Zbl 0551.53024.

[11] J. Eschenburg, Cohomology of biquotients, Manu. Math. 75 (1992) 151–166, MR 93e:57070, Zbl 0769.53029.

[12] Y. Félix, S. Halperin & J.-C. Thomas, Elliptic spaces II, Ens. Math. 39 (1993) 25–32, MR 94f:55008, Zbl 0786.55006.

[13] Y. Félix, S. Halperin & J.-C. Thomas, Rational homotopy theory, Springer, 2001, MR 2002d:55014, Zbl 0961.55002.

[14] V. Gorbatsevich & A. Onishchik, Lie transformation groups, 95–229, in ‘Lie groups and Lie algebras I’, ed. A. Onishchik, Encyclopaedia of Math. Sciences, 20, Springer, 1993, MR 1 306 739, Zbl 0781.22004.

[15] D. Gromoll & W. Meyer, An exotic sphere with nonnegative sectional curvature, Ann. Math. 100 (1974), 401–406, MR 51 #11347, Zbl 0293.53015.

[16] K. Grove & W. Ziller, Curvature and symmetry of Milnor spheres, Ann. Math. 152 (2000), 331–367, MR 2001i:53047, Zbl 0991.53016.

[17] V. Kapovitch & W. Ziller, Biquotients with singly generated rational cohomology, to appear in Geometriae Dedicata, math.DG/0210231.

[18] P. Lambrechts, The Betti numbers of the free loop space of a connected sum, J. LMS 64 (2001) 205–228, MR 2002f:55022.

[19] J. Milnor & J. Stasheff, Characteristic classes, Princeton, 1974, MR 55 #13428, Zbl 0298.57008.

[20] A. Onishchik, Inclusion relations among transitive compact transformation groups, Trudy Moskov. Mat. Obshch. 11 (1962) 199–242; Amer. Math. Soc. Translations (2) 50 (1966) 5–58, Zbl 0207.33604.

[21] A. Onishchik, Remark on invariants of groups generated by reflections, Selecta Math. Sov. 3 (1983/84), 239–241, MR 87h:20075, Zbl 0541.20028.

[22] A. Onishchik, Topology of transitive transformation groups, Johann Ambrosius Barth, Leipzig, 1994, MR 95e:57058, Zbl 0796.57001.

[23] G. Paternain, Differentiable structures with zero entropy on simply connected 4manifolds, Bol. Soc. Brasil. Mat. 31 (2000) 1–8, MR 2001c:53042.

[24] A. Petrunin & W. Tuschmann, Diffeomorphism finiteness, positive pinching, and second homotopy, Geom. Funct. Anal. 9 (1999) 736–774, MR 2000k:53031, Zbl 0941.53026.

[25] W. Singhof, On the topology of double coset manifolds, Math. Ann. 297 (1993) 133–146, MR 94k:57054, Zbl 0793.57019.

[26] B. Totaro, Complexifications of nonnegatively curved manifolds, J. Eur. Math. Soc. 5 (2003), 69–94.

[27] B. Totaro, Curvature, diameter, and quotient manifolds, Math. Res. Lett., to appear.

[28] W.-T. Wu, On Pontrjagin classes III, Amer. Math. Soc. Translations (2) 11 (1959) 155–172, MR 22 #5981, Zbl 0084.19004.

DPMMS, Wilberforce Road Cambridge CB3 0WB, England