j. differential geometry 64 (2003) 1-55

PROJECTIVE PLANES AND THEIR LOOK-ALIKES

LINUS KRAMER

Abstract

We classify all closed topological manifolds which have the same integral homology as a projective plane.

In this paper we classify manifolds which look like projective planes. More precisely, we consider 1-connected closed topological manifolds M with integral homology

H_{•}(M) ≅ ℤ^{3}.

A straight-forward application of Poincaré duality shows that for such a manifold there exists a number m ≥ 2 such that H_{k}(M) = ℤ, for k = 0, m, 2m; in particular, dim(M) = 2m is even. It follows from Adams’ Theorem on the Hopf invariant that m divides 8.

We construct a family of topological 2m-manifolds M(ξ) which are Thom spaces of certain topological ℝ^{m}-bundles (open disk bundles) ξ over the sphere 𝕊^{m}, for m = 2, 4, 8, and which we call models. This idea seems to go back to Thom and was exploited further by Shimada [53] and Eells-Kuiper [14]. A particular case is worked out in some detail in Milnor-Stasheff [46, Ch. 20]. However, these authors used vector bundles instead of ℝ^{m}-bundles. We will see that the non-linearity of ℝ^{m}-bundles yields many more manifolds than the construction by Eells-Kuiper. In [14, p. 182], the authors expressed the hope that “the given combinatorial examples form a complete set [...] for n ≠ 4”. Our results show that in dimension n = 2m = 16, their construction missed 27/28 of the (infinitely many) combinatorial and topological solutions, while in dimension n = 8, they obtained all combinatorial, but only half of the topological solutions.

Supported by a Heisenberg fellowship by the Deutsche Forschungsgemeinschaft.

Received 06/18/2002.

Next, we determine certain characteristic classes of our models, and in particular their rational Pontrjagin classes. Using these characteristic classes and Wall’s surgery sequence, we show that for m ≠ 2, every manifold which looks like a projective plane is homeomorphic to one of our models. Thus, we obtain a complete homeomorphism classification of all manifolds which are like projective planes. Furthermore, we determine which models admit DIFF (or PL) structures. The case m = 2 (so dim(M) = 4) is different, but there, we can apply Freedman’s classification of closed 4-manifolds.

We also classify the homotopy types of 1-connected Poincaré duality complexes X with H_{•}(X) ≅ ℤ^{3}. This homotopy-theoretic version of our main result (which was already proved in Eells-Kuiper [14]) is needed in the course of the homeomorphism classification; since the methods here are somewhat different from the rest of the paper, I put it in an appendix.

Main results.

Let M be a 1-connected closed topological manifold which looks like a projective plane, i.e., H_{•}(M) ≅ ℤ^{3}. Then dim(M) = 2m = 4, 8, 16.

If m = 2, then M is homeomorphic to the complex projective plane ℂP^{2} or to the Chern manifold Ch^{4}. These two manifolds are topologically distinguished by their Kirby-Siebenmann numbers ks[M] ∈ ℤ/2; the Chern manifold (ks[Ch^{4}] ≠ 0) admits no DIFF structure (this case is due to Freedman [19]).

If m = 4, then M is homeomorphic to one of our models M(ξ). Topologically, it is determined by the Pontrjagin number p_{4}^{2}[M] ∈ {2(1+ 2t)^{2} | t ∈ ℤ} and the Kirby-Siebenmann number ks^{2}[M] ∈ ℤ/2. These data also determine the oriented bordism class of M, so no two models are equivalent under oriented bordism. The manifold admits a PL structure (unique up to isotopy) if and only if ks^{2}[M] = 0.

If m = 8, then M is homeomorphic to one of our models M(ξ). Topologically, it is determined by the Pontrjagin number

p_{8}^{2}[M] ∈ \{\frac{36}{49}(1 + 2t)^{2} | t ∈ ℤ\}

and a characteristic number \frac{7}{6}p_{8}κ[M] ∈ ℤ/4, determined by the integral characteristic class \frac{7}{6}p_{8}(M) and a certain 8-dimensional PL characteristic class κ with ℤ/4-coefficients. These data also determine the oriented bordism class of M, so no two models are equivalent under oriented bordism. These manifolds admit a PL structure (unique up to isotopy).

We determine also which of these manifolds admit a DIFF structure, and determine the homotopy type in terms of the characteristic classes. See Sections 7 and 8 for more detailed statements. As a by-product of our proof, we obtain an explicit classification of ℝ^{m}-bundles over 𝕊^{m} in terms of characteristic classes, for m = 2, 4, 8.

* * *

Topological geometry plays no rôle in this paper. However, the motivation to write it came from a long-standing open problem in topological geometry:

(∗) What are the possible homeomorphism types of the point spaces of compact projective planes (in the sense of Salzmann [52])?

The point space P of a compact projective plane is always the Thom space of a locally compact fiber bundle, see Salzmann et al. [52] Ch. 5, in particular 51.23. (Problem (∗) should not be confused with the geometric problem of classifying all compact projective planes with large automorphism groups which was solved by Salzmann and his school [52].)

Now (∗) turns out to be a difficult problem. The present state of affairs is as follows, see [52]. Let P be the point space of a compact projective plane. If the covering dimension of P is dim(P) = 0, then P is either finite or homeomorphic to the Cantor set {0, 1}^{ℕ}. If 1 ≤ dim(P) ≤ 4, then P ≅ ℝP^{2} or P ≅ ℂP^{2}; this was proved by Salzmann and Breitsprecher already in the late 60s [7] (surprisingly, this did not require results about 4-manifolds). The proof depends on a result by Borsuk about low-dimensional ANRs and on Kneser’s Theorem SO(2) ≃ STOP(2). In (finite) dimensions bigger than 4, Löwen [40] applied sheaftheoretic cohomology to the problem. Using a beautiful local-to-global argument, he proved that P is an m − 1-connected Poincaré duality complex and an integral 2m-dimensional ENR manifold with H_{•}(P) ≅ ℤ^{3}, and that m = 2, 4, 8.

So the topological problem (∗) is reduced to the following steps:

(1) Prove that the topological dimension dim(P) is finite.

(2) Assuming that dim(P) < ∞, prove that P is a manifold (and not just an integral ENR manifold).

(3) Assuming that P is a manifold, determine its homeomorphism type.

Each step seems to be difficult. Under the additional assumption that the compact projective plane is smooth (in the sense of [52]: the geometric operations are smooth maps), a complete homeomorphism classification (based on characteristic classes) of the point spaces was carried out in [36]. Buchanan [9] determined the homeomorphism types of the point spaces of compact projective planes coordinatized by real division algebras by a direct homotopy-theoretic argument (note that besides the classical alternative division algebras ℝ, ℂ, ℍ, and 𝕆, there exists a continuum of other real division algebras). In both cases, the homeomorphism types of the point spaces turn out to be the classical ones, ℝP^{2}, ℂP^{2}, ℍP^{2}, or 𝕆P^{2}. My hope is that the results in this paper, together with Knarr’s Embedding Theorem [35] and the result in [37] will eventually lead to a solution of (3).

* * *

I have tried to make the paper self-contained and accessible to nonexperts. There is necessarily a certain overlap with the paper by Eells-Kuiper [14]. My aim was to give complete proofs for all steps of the classification, starting only from general facts about bundles and manifolds. Thus, the reader is not assumed to be familiar with [14] (although this fundamental paper is certainly to be recommended).

Standing assumptions. An n-manifold (without boundary) is a metrizable, second countable space which is locally homeomorphic to ℝ^{n}. Throughout, all maps are assumed to be continuous. Except for the appendix, maps and homotopies are not required to preserve base points, unless the contrary is stated explicitly. Thus [X; Y] denotes the set of all free homotopy classes of maps from X to Y. If X, Y are wellpointed spaces, and if Y is 0-connected, then the fundamental group π_{1}(Y) acts on the set [X; Y]_{0} of based homotopy classes; the set [X; Y] of all free homotopy classes can be identified with the orbit set of this action, see Whitehead [66] Ch. III.1. If Y is an H-space (or if Y is 1-connected) this action is trivial, so [X; Y] = [X; Y]_{0}.

Acknowledgements. I am indebted to Stephan Stolz and Michael Weiss for sharing some of their insights.

1. Preliminaries on bundles

Fiber bundles, microbundles, fibrations, and Thom spaces play a prominent rôle in this paper, so we briefly recall the relevant notions. We refer to Holm [26], Milnor [43], Dold [11], and to the books by Kirby-Siebenmann [33], Rudyak [50], and Husemoller [30].

1.1. A bundle ϕ = (E, B, p) over a space B is a map E ^{p} B. The class of all bundles over B forms in an obvious way a category whose morphisms ϕ ^{f} ϕ′ are commutative diagrams

[figure: E′; f; E; B; p′; = = = =; B.; p]

An isomorphism in this category is called an equivalence of bundles and denoted ϕ ≅ ϕ′; in the diagram above, f is an equivalence if and only if f is a homeomorphism. The categorical product of two bundles ϕ, ϕ′ is the Whitney sum ϕ ⊕ ϕ′; its total space is E ⊕ E′ = {(e, e′) ∈ E × E′ | p(e) = p′(e′)}, with the obvious bundle projection. A homotopy between two morphisms f_{0}, f_{1} : ϕ′ [figure] ϕ or homotopy over B is a homotopy E′ × [0, 1] E with the property that the diagram

[figure: E′; f_{t}; E; B; = = = = =; B]

commutes for all t ∈ [0, 1]. A morphism f is called a fiber homotopy equivalence if it has a homotopy inverse bundle map g, i.e., if fg and gf are homotopic over B to the respective identity maps; in this case we write ϕ ≃ ϕ′. A section of a bundle ϕ = (E, B, p) is a morphism s from the identity bundle (B, B, id_{B}) to ϕ,

[figure: E; s; B; = = = =; B.; p]

and we call (E, B, p, s) a sectioned bundle.

A map B′ ^{g} B induces a contravariant functor g^{∗} which assigns to every bundle ϕ = (E, B, p) the pull-back bundle g^{∗}ϕ = (g^{∗}E, B′, p′), with g^{∗}E = {(e, b′) ∈ E × B′ | p(e) = g(b′)} and p′(e, b′) = b′. If g is a homeomorphism, then g^{∗} is an equivalence of categories; in this case, two bundles ϕ and ϕ′ are called weakly equivalent if ϕ′ is equivalent to g^{∗}ϕ, in other words, if there are homeomorphisms

[figure: E′; f ; ≅; E; B′; p′; g ; ≅; B; p]

commuting with the bundle projections; such a weak equivalence is denoted ϕ ≅_{g} ϕ′.

1.2. A bundle is called a fibration if the homotopy extension problem

X × \{0\} \overline{f} E

.......

[figure: X × [0, 1]; ∩; g; B; p]

has a solution for every space X. We call a fibration n-spherical if every fiber E_{b} = p^{−1}(b) has the homotopy type of an n-sphere.

For a subspace A ⊆ B, we have the restriction ϕ|_{A} = (E_{A} = p^{−1}(A), A, p|_{E_{A}}) of the bundle ϕ.

Definition 1.3. A bundle is called a fiber bundle with typical fiber F if every b ∈ B has an open neighborhood U such that the restriction ϕ|_{U} is equivalent to the product (or trivial) bundle (F × U, U, pr_{2})

[figure: F × U; ≅; E_{U}; ⊂; E; U; pr_{2}; = = = = = =; U; ⊂; B;]

such a local trivialization is also called a coordinate chart for the bundle. If in addition a base point is fixed in the fiber F, one obtains in an obvious way a sectioned fiber bundle.

For technical reasons, it is often convenient to consider numerable fiber bundles. For example, every numerable fiber bundle is automatically a fibration, see Spanier [55] Ch. 2.7 Theorem 12.

Definition 1.4. A locally finite covering {V_{i} | i ∈ I} of B by open sets is called numerable if there exist maps f_{i} : B \overline{}[0, 1] with f_{i}^{−1}((0, 1]) = V_{i}, such that ∑_{i∈I} f_{i} = 1. A fiber bundle is called numerable if there exists a numerable covering of B by coordinate charts. In our setting, most base spaces will be paracompact, so fiber bundles are automatically numerable.

Definition 1.5. An n-sphere bundle is a numerable fiber bundle with 𝕊^{n} as typical fiber. An ℝ^{n}-bundle is a sectioned numerable bundle with (ℝ^{n}, 0) as typical fiber; the section is denoted s_{0} and called the zerosection. The trivial ℝ^{n}-bundle (over any space) will be denoted \underline{ℝ}^{n}; its total space is E = ℝ^{n}×B, with p = pr_{2}. An n-dimensional vector bundle is an ℝ^{n}-bundle which carries in addition a real vector space structure on each fiber which is compatible with the given coordinate charts. Two ℝ^{n}-bundles (or vector bundles) ξ, ξ′ are called stably equivalent if there is an equivalence ξ ⊕ \underline{ℝ}^{k} ≅ ξ′ ⊕ \underline{ℝ}^{k′}, for some k, k′ ≥ 0.

A crucial property of ℝ^{n}-bundles is the following homotopy property.

Lemma 1.6. Let ξ be an ℝ^{n} bundle over B. If g_{0}, g_{1} : B′ [figure] B are homotopic, then there is an equivalence g_{0}^{∗}ξ ≅ g_{1}^{∗}ξ.

Proof. See Holm [26] Lemma 1.5. q.e.d.

1.7. In an ℝ^{n}-bundle ξ = (E, B, ps_{0}), the zero-section s_{0} is a homotopy inverse to the bundle projection p, and s(B) is a strong deformation retract of the total space E, see Holm [26] Theorem 3.6. In particular, the section s_{0} : B E is a cofibration. It follows that the quotient E/s_{0}(B) is contractible.

Definition 1.8. From each ℝ^{n}-bundle ξ, one obtains an n-sphere bundle ^{s}ξ by compactifying each fiber of ξ. The resulting bundle ^{s}ξ has two sections, the zero-section s_{0} and the section s_{∞} corresponding to the new points added in the fibers. Let E denote the total space of ξ, and put E_{0} = E \ s_{0}(B), the total space with the zero-section removed. Finally, let^{u}E = E_{0}∪s_{∞}(B). Then clearly,^{u}ξ = (^{u}E, B,^{u} p, s_{∞}) is again an ℝ^{n}-bundle (called the upside down bundle in [37]), and E_{0} B is a numerable fiber bundle with ℝ^{n} \ 0 as typical fiber. We call E_{0} B the spherical fibration corresponding to the ℝ^{n}-bundle ξ. The Thom space M(ξ) of an ℝ^{n}-bundle ξ is the quotient

M(ξ) = E ∪ s_{∞}(B)/s_{∞}(B) = E ∪ {o}.

We denote the base point (the tip) of M(ξ) by o. If B is compact, then M(ξ) = E ∪ {o} is the same as the one-point compactification of E.

1.9. By the previous remarks, o is a strong deformation retract of E_{0} ∪ {o}. In particular, there is a natural (excision) isomorphism H^{•}(E, E_{0}) ≅ H^{•}(M(ξ), o).

We need also the concept of a microbundle, see Milnor [43] and Holm [26].

Definition 1.10. An n-microbundle 𝔵 = (E, B, p, s) is a sectioned bundle, subject to the following condition: for every b ∈ B there exists an open neighborhood U of b, an open subset V ⊆ E_{U} = p^{−1}(U) containing s(U) and a section-preserving homeomorphism h : U × ℝ^{n} V such that the diagram

[figure: ℝ^{n} × U; h; V; ⊂; E_{U}; ⊂; E; U; pr_{2}; = = = = = =; U; = = = = =; U; ⊂; B;]

commutes. The difference between a microbundle and an ℝ^{n}-bundle is that h need not be surjective onto E_{U}. Similarly as for ℝ^{n}-bundles, we require the existence of a numerable covering of B by such local charts.

Clearly, every ℝ^{n}-bundle is an n-microbundle. It is also clear that there exist microbundles which are not fiber bundles. A particularly important example is the tangent microbundle 𝔱M of a manifold M: here, E = M × M, the bundle projection is pr_{2} and the section is the diagonal map, s(x) = (x, x), see Milnor [43] Lemma 2.1.

The Kister-Mazur Theorem (see Theorem 1.11 below) says that microbundles are in a sense equivalent to ℝ^{n}-bundles, a fact which is not obvious at all. If E′ ⊆ E is a neighborhood of s(B), then it is not difficult to see that E′ B is again a microbundle 𝔵′ contained in 𝔵. Two microbundles 𝔵_{1}, 𝔵_{2} over the same base B are called micro-equivalent if they contain microbundles 𝔵′_{1}, 𝔵′_{2} which are equivalent as bundles (this is also sometimes called a micro-isomorphism or an isomorphism germ).

In the case of numerable microbundles one has to be careful: a microbundle contained in a numerable microbundle need a priori not be numerable.

Theorem 1.11 (Kister-Mazur). Let 𝔵 be a numerable n-micro-bundle. Then there exists a numerable microbundle 𝔵′ contained in 𝔵 which is an ℝ^{n}-bundle, and 𝔵′ is unique up to equivalence.

Proof. See Holm [26] Theorem 3.3. q.e.d.

In particular, the tangent microbundle 𝔱M of any (metrizable) nmanifold M contains an ℝ^{n}-bundle, unique up to equivalence. We will call this ℝ^{n}-bundle τM (and any bundle equivalent to it) the tangent bundle of M. If M happens to be a smooth manifold, one can show that τM is equivalent to the smooth tangent bundle T M, see Milnor [43] Theorem 2.2.

If ξ is an ℝ^{n}-bundle over a manifold B, then the total space E is clearly a manifold, and the zero-section s_{0}(B) is a submanifold with normal (micro) bundle ξ, see Milnor [43] Sec. 5. We require the following splitting result.

Proposition 1.12. There is an equivalence

s_{0}^{∗}τE ≅ τB ⊕ ξ.

Proof. This follows from Milnor [43] Theorem 5.9, combined with the Kister-Mazur Theorem 1.11 above. q.e.d.

2. Constructing the models

In this section we construct a family of manifolds as Thom spaces of ℝ^{m}-bundles over the sphere 𝕊^{m}, for m = 2, 4, 8, which we call models. We begin with some general remarks about Thom spaces of ℝ^{m}-bundles over 𝕊^{m}. We fix a generator [𝕊^{m}] ∈ H_{m}(𝕊^{m}). Let ξ be an ℝ^{m}-bundle over 𝕊^{m}, for m ≥ 2, with total space E, and let E_{0} = E \ s_{0}(𝕊^{m}) denote the total space with the zero-section removed. Since 𝕊^{m} is 1connected, the bundle ξ is orientable, and we may choose an orientation class u(ξ) ∈ H^{m}(E, E_{0}), see Spanier [55] Ch. 5.7 Corollary 20. The image e(ξ) = s_{0}^{•}(u(ξ)|_{E}) of u(ξ) in H^{m}(𝕊^{m}) is the Euler class of ξ. We call the integer

|e| = |⟨e(ξ), [𝕊^{m}]⟩|

the absolute Euler number of ξ; it is independent of the choice of [𝕊^{m}] and of u(ξ). Let M(ξ) denote the Thom space of ξ, and let

Φ : H^{•}(B) ^{≅} H^{•+m}(E, E_{0}), Φ(v) = p^{•}(v)⌣u(ξ)

denote the Gysin-Thom isomorphism, see Spanier [55] Ch. 5.7 Theorem 10. By 1.9, this yields — via excision — isomorphisms

H^{•}(𝕊^{m}) ≅ H^{•+m}(E, E_{0}) ≅ H^{•+m}(M(ξ), o).

Let y_{m}, y_{2m} be generators for the infinite cyclic groups H^{m}(M(ξ)) and H^{2m}(M(ξ)), respectively.

Lemma 2.1. In the cohomology ring H^{•}(M(ξ)), we have the relation y_{m}^{2} = ±|e|y_{2m}.

Proof. Let x ∈ H^{m}(𝕊^{m}) be the generator dual to [𝕊^{m}] and let Φ denote the Gysin-Thom isomorphism. Thus Φ(1) = u(ξ) = ±y_{m} and Φ(x) = ±y_{2m}. Let e(ξ) = εx, for ε ∈ ℤ (so |e| = |ε|). Then Φ(e(ξ)) = εΦ(x) = u(ξ)⌣u(ξ), since p^{•}(e(ξ)) = u(ξ)|_{E}. q.e.d.

2.2. If m is odd, then u(ξ) ⌣ u(ξ) = 0, so e(ξ) = 0. Therefore, |e| = 0 if m is odd.

Proposition 2.3. If M(ξ) is a manifold, then |e| = 1 and m is even and divides 8. Moreover, H^{•}(M(ξ); R) ≅ R[y_{m}]/(y_{m}^{3}) for any commutative ring R.

Proof. If M(ξ) is a manifold with fundamental class μ, then Poincaré duality implies that the map

H^{m}(M(ξ)) ⊗ H^{m}(M(ξ))  ℤ,  u ⊗ v ⟼ ⟨u⌣v, μ⟩

is a duality pairing, so |e| = 1 and m is even. Thus H^{•}(M(ξ); R) ≅ R[y_{m}]/(y_{m}^{3}) for any commutative ring R. Since M(ξ) is a manifold, it is an ANR, see Hanner [21] Theorem 3.3 or Hu [28] p. 98 and thus homotopy equivalent to a CW-complex X, see Weber [65] p. 218; by standard obstruction theory, X ≃ 𝕊^{m} ∪ e^{2m} is homotopy equivalent to a 2-cell complex, see Wall [63] Proposition 4.1. By Adams-Atiyah [2] Theorem A, this implies that m = 2, 4, 8. More details can be found in the appendix. q.e.d.

The exact homotopy sequence of the m − 1-spherical fibration E_{0} 𝕊^{m} shows that π_{1}(E_{0}) is abelian, and that E_{0} is m − 2-connected.

For |e| = 1, the Gysin sequence

H_{k}(E_{0})  H_{k}(𝕊^{m}) ^{e(ξ)⌢} H_{k−m}(𝕊^{m})  H_{k−1}(E_{0})

breaks down into isomorphisms H_{0}(E_{0}) ≅ H_{0}(𝕊^{m}) and H_{m}(𝕊^{m}) ≅ H_{2m−1}(E_{0}), and all other homology groups of E_{0} vanish. A repeated application of the Hurewicz isomorphism shows that E_{0} is 2(m − 1)- connected, with π_{2m−1}(E_{0}) ≅ H_{2m−1}(E_{0}) ≅ ℤ. Thus

H_{k}(E_{0} ∪ \{o\}, E_{0}) ≅ H̃_{k−1}(E_{0}) ≅ \begin{cases}ℤ & for k = 2m \\ 0 & else, \end{cases}

because E_{0} ∪ {o} is contractible. In other words, M(ξ) has for |e| = 1 the same local homology groups at o as ℝ^{2m}.

Recall that, due to the embeddability of second countable finite dimensional metric spaces, a locally compact finite dimensional second countable ANR (absolute neighborhood retract for the class of metric spaces see Hu [28]) is exactly the same as an ENR (Euclidean neighborhood retract, see Hurewicz-Wallman [29] Ch. V, Engelking [17] Theorem 1.11.4, and Dold [13] Ch. IV.8).

Lemma 2.4. For |e| = 1, the Thom space M(ξ) = E ∪ {o} is an integral ENR 2m-manifold, i.e., an ENR (Euclidean neighborhood retract) which has the same local homology groups as ℝ^{2m}.

Proof. The space E ∪ s_{∞}(𝕊^{m}) is a 2m-manifold (and in particular an ENR), and M(ξ) = E ∪ {o} = E ∪ s_{∞}(𝕊^{m})/s_{∞}(𝕊^{m}) is a quotient of an ENR (the manifold E ∪ s_{∞}(𝕊^{m})) by a compact ENR subspace (the m-sphere s_{∞}(𝕊^{m})). Such a quotient is again an ENR, see Hanner [21] Theorem 8.2 or Hu [28] Ch. IV. The local homology groups at o were determined above; every point in E has a locally Euclidean neighborhood and thus the same local homology groups as ℝ^{2m}. q.e.d.

Our next aim is to show that M(ξ) is in fact a manifold. Since M(ξ) = E ∪ {o} and E is a manifold, the only point which we have to consider in detail is o. First, we prove that o is 2(m−1)-LC in M(ξ), i.e., that every open neighborhood V of o contains an open neighborhood V′ of o such that for k ≤ 2(m−1), every map 𝕊^{k} V′ \{o} is homotopic in V \ {o} to a constant map. Clearly, we are done if we can show that V′ \ {o} is 2(m − 1)-connected.

Lemma 2.5. The space M(ξ) is 2(m − 1)-LC at o if |e| = 1.

Proof. Let ^{u}E = E_{0} ∪ s_{∞}(𝕊^{m}) denote the upside-down bundle obtained from ξ (see Definition 1.8). Then ^{u}E 𝕊^{m} is an ℝ^{m} bundle and in particular a microbundle (see Definition 1.10). Let V be an open neighborhood of o in M(ξ), and let f denote the map uE E_{0} ∪ {o} ⊆ M(ξ) which collapses the s_{∞}-section to the point o. Then U = f^{−1}(V) is an open neighborhood of s_{∞}(𝕊^{m}) in the upsidedown bundle ^{u}E. By the Kister-Mazur Theorem 1.11, there exists an open neighborhood U′ of s_{∞}(𝕊^{m}) in U with ^{u}E ⊇ U ⊇ U′, such that U′ 𝕊^{m} is an ℝ^{m}-bundle equivalent to ^{u}E 𝕊^{m}. In particular, U′ \s_{∞}(𝕊^{m}) ≅ E_{0} is 2(m−1)-connected. Now we put V′ = f(U′). q.e.d.

Corollary 2.6. For |e| = 1, the Thom space M(ξ) is a 1-connected closed 2m-manifold.

Proof. The space M(ξ) \ {o} is a 2m-manifold, and M(ξ) is 1-LC at o. Thus, o has an open neighborhood homeomorphic to ℝ^{2m}; for m = 2, this follows from Freedman-Quinn [20] Theorem 9.3A (and also from Kneser’s Theorem TOP(2) ≃ O(2), see Theorem 6.4 below), and for m = 4, 8 from Quinn [49] Theorem 3.4.1. Van Kampen’s Theorem, applied to the diagram

E_{0}
[figure]
E  E_{0} ∪ {o}
[figure]
M(ξ)

shows that π_{1}(M(ξ)) = 0, because E, E_{0} ∪{o}, and E_{0} are 1-connected. q.e.d.

Definition 2.7. The manifolds M(ξ) obtained in this way as Thom spaces of ℝ^{m}-bundles with |e| = 1 will be called models.

The same argument as above shows that E/s_{0}(𝕊^{m}) is a manifold, and so S = M(ξ)/s_{0}(𝕊^{m}) is a manifold, too. Similarly as above, Van Kampen’s Theorem shows that S is 1-connected. As E_{0} has the same homology as 𝕊^{2m−1}, the exact homology sequence of the pair (E, E_{0}) shows that the composite 𝕊^{m} ^{s_{0}} E (E, E_{0}) is an isomorphism in m-dimensional homology. Thus H̃_{•}(S) ≅ H_{•}(M(ξ), s_{0}(𝕊^{m})) (here we use that s_{0} : 𝕊^{m} M(ξ) is a cofibration). Therefore, S is a 1-connected homology 2m-sphere, and thus, by the proof of the generalized Poincaré conjecture in higher dimensions, a sphere (in dimension 4, see Freedman [19], and in higher dimension Smale [54] and Newman [47]). In particular, E_{0} ∪ {o} ≅ ℝ^{2m} is an open cell. Thus, M(ξ) is a compactification of an open 2m-cell by an m-sphere.

Proposition 2.8. Each model M(ξ) can be decomposed as

M(ξ) = X ˙∪ U,

with X = s_{0}(𝕊^{m}) homeomorphic to 𝕊^{m}, and U = E_{0} ∪ {o} open, dense, and homeomorphic to ℝ^{2m}.

3. Homeomorphisms between different models

In the last section, we constructed for every oriented ℝ^{m}-bundle ξ over 𝕊^{m} with absolute Euler number |e| = |⟨e(ξ), [𝕊^{m}]⟩| = 1 a manifold M(ξ). In this section, we determine under which conditions there are homeomorphisms M(ξ) ≅ M(ξ′) between different models. Clearly, a weak bundle equivalence ξ ≅_{g} ξ′ induces a homeomorphism M(ξ) ≅ M(ξ′) between the Thom spaces. We will see that this is in fact the only possibility. In the first part of this section, we assume only that m is even; in the second part, we return to the special case of our models where m = 2, 4, 8 and |e| = 1.

Let ξ be an ℝ^{m}-bundle over 𝕊^{m}, with absolute Euler number |e|, for m ≥ 2 even. Let X = s_{0}(𝕊^{m}) ⊆ E. By Proposition 1.12, the tangent bundle τE of the manifold E splits along X as a sum of a horizontal and a vertical bundle. Since τ𝕊^{m} ⊕ \underline{ℝ} ≅ \underline{ℝ}^{m+1}, we have the following result.

Lemma 3.1. Let τE denote the topological tangent bundle of E, and let X = s_{0}(𝕊^{m}). Then X ⊆ E is an embedded submanifold with normal bundle weakly equivalent to ξ and

ξ ⊕ τ𝕊^{m} ≅ s_{0}^{∗}τE.

In particular, ξ and τE|_{X} are weakly stably equivalent,

ξ ⊕ \underline{ℝ}^{m+1} ≅ s_{0}^{∗}τE ⊕ \underline{ℝ}.

Suppose now that there is a homeomorphism E f ; ≅ E′ of total

spaces of ℝ^{m} bundles ξ, ξ′ over 𝕊^{m}, for m ≥ 2 even. Both s_{0} and f^{−1}s′_{0} represent generators of π_{m}(E) ≅ ℤ. Thus, there exists a homeomorphism g : 𝕊^{m} 𝕊^{m} of degree ±1 such that the diagram

[figure: E; f ; ≅; E′; 𝕊^{m}; s_{0}; g ; ≅; 𝕊^{m}; s′_{0}]

is homotopy commutative. By Lemma 1.6, there is an equivalence s_{0}^{∗}τE ≅ g^{∗}(s′_{0})^{∗}τE′, whence ξ ⊕ \underline{ℝ}^{m+1} ≅ g^{∗}ξ′ ⊕ \underline{ℝ}^{m+1}. In other words, the bundles ξ and ξ′ are weakly stably equivalent.

Lemma 3.2. Suppose that there is a homeomorphism of Thom spaces M(ξ) ≅ M(ξ′). Then there is a homeomorphism between the total spaces E ≅ E′, and the bundles ξ, ξ′ have the same absolute Euler number.

Proof. By Lemma 2.1, the absolute Euler number can be seen from the cohomology ring of M(ξ), so |e| = |e′|. If M(ξ) and M(ξ′) are manifolds, then they are homogeneous and the existence of a homeomorphism M(ξ) ≅ M(ξ′) implies the existence of a homeomorphism E ≅ E′. If M(ξ) and M(ξ′) are not manifolds, then a homeomorphism maps the unique non-manifold point o of M(ξ) onto the unique nonmanifold point o′ of M(ξ′), and so it maps E onto E′. q.e.d.

The proof of the next proposition involves classifying spaces, so we postpone it to 6.3.

Proposition 3.3. Let ξ, ξ′ be ℝ^{m}-bundles over 𝕊^{m}, for m ≥ 2 even. Suppose that there is a stable equivalence ξ ⊕ \underline{ℝ}^{k} ≅ ξ′ ⊕ \underline{ℝ}^{k}, and that the absolute Euler numbers of ξ and ξ′ are equal, |e| = |e′|. Then there is an equivalence ξ ≅ ξ′.

Combining these results, we obtain a complete homeomorphism classification of the Thom spaces M(ξ), for m ≥ 2 even, in terms of bundles.

Proposition 3.4. Let m ≥ 2 be even, let ξ, ξ′ be ℝ^{m}-bundles over 𝕊^{m}. If there is a homeomorphism between the Thom spaces M(ξ) ≅ M(ξ′), then ξ and ξ′ are weakly equivalent.

Proof. By Lemma 3.2 above, the total spaces E, E′ are homeomorphic, and |e| = |e′|. The remarks at the begin of this section show that there is a weak equivalence between ξ⊕\underline{ℝ}^{m+1} and ξ′⊕\underline{ℝ}^{m+1}, induced by a homeomorphism g : 𝕊^{m} 𝕊^{m}. If deg(g) = 1, then g is homotopic to the identity, whence ξ ⊕ \underline{ℝ}^{m+1} ≅ ξ′ ⊕ \underline{ℝ}^{m+1}. By Proposition 3.3, this implies that there is an equivalence ξ ≅ ξ′. Otherwise, deg(g) = −1 and we put ξ′′ = g^{∗}ξ′. Then we have an equivalence ξ ⊕ \underline{ℝ}^{m+1} ≅ ξ′′ ⊕ \underline{ℝ}^{m+1}, and, again by Proposition 3.3, an equivalence ξ ≅ ξ′′. But ξ′ is weakly equivalent to ξ′′. q.e.d.

Thus we have reduced the homeomorphism classification of our models to a classification of ℝ^{m}-bundles over 𝕊^{m}. For the specific values m = 2, 4, 8, this classification will be carried out in the next section. We end this section with some simple remarks about characteristic classes of our models. Each model M(ξ) has a distinguished orientation: if y_{m} is any generator of H^{m}(M(ξ)), then y_{m}^{2} is a generator of H^{2m}(M(ξ)) which does not depend on the choice of y_{m}. So we choose for our models the fundamental class [M(ξ)] ∈ H^{2m}(M(ξ)) in such a way that

⟨y_{m}^{2}, [M(ξ)]⟩ = 1.

Obviously, the Euler characteristic of any model M(ξ) is

χ_{M(ξ)} = 3.

Note also that any homeomorphism c : 𝕊^{m} 𝕊^{m} of degree −1 induces a homeomorphism f_{c} of M(ξ) with the property that f_{c}^{•}y_{m} = −y_{m}. Thus every graded automorphism of the cohomology ring H^{•}(M(ξ)) ≅ ℤ[y_{m}]/(y_{m}^{3}) is induced by a homeomorphism.

Recall that the Wu classes v_{i} ∈ H^{i}(M; ℤ/2) of a closed manifold M are defined by ⟨v_{i} ⌣ x, [M]⟩ = ⟨Sq^{i}x, [M]⟩.

Lemma 3.5. The total Stiefel-Whitney class of any model M(ξ) is given by

w(M(ξ)) = 1 + y_{m} + y_{m}^{2},

where y_{m} ∈ H^{m}(M(ξ); ℤ/2) is a generator. Thus the minimal codimension for an embedding of M(ξ) in 𝕊^{2m+k} or ℝ^{2m+k} is k = m + 1.

Proof. We have Sq^{m} y_{m} = y_{m}^{2}, so the total Wu class of M(ξ) is v = 1 + y_{m} + y_{m}^{2}, and the total Stiefel-Whitney class is w(M(ξ)) = Sq v = 1 + y_{m} + y_{m}^{2} (see Spanier [55] Ch. 6.10 Theorem 7 and 6.10 8). The non-embedding result follows as in Spanier [55] Ch. 6.10 24. q.e.d.

Recall from 1.7 that the composite 𝕊^{m} ^{s_{0}} X ⊆ E is a homotopy equivalence. Since E_{0} is 2(m − 1)-connected, the exact homology sequence of the pair (E, E_{0}) shows that H_{m}(E) H_{m}(E, E_{0}) is an isomorphism.

Lemma 3.6. The map s_{0} : 𝕊^{m} M(ξ) induces an isomorphism on the homotopy and (co)homology groups up to (and including) dimension m.

Proof. Since M(ξ) is 1-connected, the claim on the homotopy groups follows from the corresponding result for the homology groups by the Hurewicz isomorphism. q.e.d.

3.7. In particular, if γ(M) is a stable m-dimensional characteristic class of the tangent bundle of M(ξ) then s_{0}^{•}(γ(M)) = γ(ξ). For m = 4, 8, this applies in particular to the m-dimensional rational Pontrjagin class p_{m} of (the tangent bundle of) M(ξ),

p_{m}(ξ) = s_{0}^{•}p_{m}(M(ξ)),

so the m-dimensional Pontrjagin class of ξ determines the m-dimensi-onal Pontrjagin class of M(ξ). A similar result holds for the exotic classes, the Kirby-Siebenmann class ks and the class κ of M, which are constructed in 4.13. This will be used in the Section 4.

4. Characteristic classes

To get further, we need some results about classifying spaces. We refer to the books by Milnor-Stasheff [46], Kirby-Siebenmann [33], Madsen-Milgram [41] and to Ch. IV in Rudyak [50]. We denote the orthogonal group by O(n), and by TOP(n) the group of all base-point preserving homeomorphisms of ℝ^{n}. The corresponding classifying spaces are BO(n) and BTOP(n). If X is any space, then the set of free homotopy classes [X; BO(n)] is in one to one correspondence with the equivalence classes of numerable n-dimensional vector bundles over X. Similarly, BTOP(n) classifies numerable ℝ^{n}-bundles over X. Taking the limit n ≫ 1 one obtains stable versions O, TOP of these groups; there are corresponding classifying spaces BO and BTOP which classify ℝ^{n}-bundles up to stable equivalence. We need two more classifying spaces. The space BPL(n) classifies ℝ^{n}-bundles (over simplicial complexes) which admit PL (piecewise linear) coordinate charts. To us, the main purpose of BPL(n) and its stable version BPL will be the fact that it lies somewhat in the middle between BTOP and BO. Finally, let G(n) denote the semigroup of all self-equivalences of the sphere 𝕊^{n−1}. The corresponding classifying space BG(n) classifies n − 1-spherical fibrations up to fiber homotopy equivalence. There are 1-connected coverings of these spaces which classify oriented bundles and fibrations: for example, the classifying space BSO(n) classifies n-dimensional oriented vector bundles. For k ≥ 0 there are ladders of maps

[figure: BO(n); BO(n + k); BO]
[figure]
[figure: BSO(n); BSO(n + k); BSO]
[figure: BPL(n); BPL(n + k); BPL]
[figure]
[figure: BSPL(n); BSPL(n + k); BSPL]
[figure: BTOP(n); BTOP(n + k); BTOP]
[figure]
[figure: BSTOP(n); BSTOP(n + k); BSTOP]
[figure: BG(n); BG(n + k); BG]
[figure]
[figure: BSG(n); BSG(n + k); BSG]

such that the diagram commutes (at least up to homotopy). The horizontal arrows correspond to the process of stabilization, i.e., if f : X BSTOP(n) classifies ξ, then the composite

X  BSTOP(n)  BSTOP(n + k)

classifies ξ⊕\underline{ℝ}^{k}. The vertical and the slanted arrows are ’forgetful’: they forget the vector space structure, the PL structure, and the fiber bundle structure, respectively, and the slanted arrows forget the orientation. For these results, see e.g., Rudyak [50] Ch. IV.

By a well-known construction, every continuous map f : X Y between topological spaces can be converted into a fibration f′ : P_{f}

Y, with X ≃ P_{f}, see Spanier [55] Ch. 2.8 Theorem 9. If Y is path-connected, then all fibers of f′ have the same homotopy type, and it makes sense to speak about the homotopy fiber of the map f. Let BH BG be one of the maps in the diagram above. The homotopy fiber of this map is denoted by G/H. One can show that the homotopy fiber of the map BO(n) BO(n + k) is homotopy equivalent to the Stiefel manifold O(n + k)/O(n), so this terminology fits together with the standard Lie group terminology.

From the homotopy viewpoint, the stable classifying spaces are much easier to understand. This is partly due to the fact that they are Hspaces (the Whitney sum of bundles is the multiplication). The homotopy groups of BO are known by Bott periodicity, BO × ℤ ≃ Ω^{8}BO, see Bott [4] [5], i.e.,

π_{k}(BO × ℤ) ≅ \begin{cases}ℤ & for k ≡ 0 (mod 4) \\ ℤ/2 & for k ≡ 1, 2 (mod 8) \\ 0 & else. \end{cases}

The homotopy groups of BG correspond to the stable homotopy groups of spheres, see, e.g., Milnor [44] §2 and Madsen-Milgram [41] Ch. 3,

π_{k+1}(BG) ≅ lim_{n→∞} π_{k+n}(𝕊^{n}) = π_{k}^{s}(𝕊^{0})

which are known in low dimensions, see, e.g., Toda [60] Ch. XIV, Hu [27] pp. 328–332, or Fomenko-Fuchs-Gutenmacher [18] pp. 300–301. For the other homotopy fibers, we use the following results which are obtained from surgery theory.

Theorem 4.1. The homotopy groups of G/TOP are given by the periodicity G/TOP×ℤ ≃ Ω^{4}(G/TOP), see Kirby-Siebenmann [33] p. 327 (the ℤ-factor is forgotten there), and

π_{k}(G/TOP × ℤ) ≅ \begin{cases}ℤ & for k ≡ 0 (mod 4) \\ ℤ/2 & for k ≡ 2 (mod 4) \\ 0 & else, \end{cases}

see also Madsen-Milgram [41] Ch. 2.

Finally, we use the following result (see Kirby-Siebenmann [33, p. 200]).

Theorem 4.2. The homotopy groups of TOP/O and TOP/PL are finite in all dimensions. If i ≥ 5, then π_{i}(TOP/O) is isomorphic to the Kervaire-Milnor group Θ_{i} of DIFF structures on 𝕊^{i} (see Kirby-Siebenmann [33] p. 200, 251), and TOP/PL is an Eilenberg-MacLane space of type K(ℤ/2, 3).

It follows from Serre’s C-theory that

H̃^{•}(TOP/O; ℚ) ≅ H̃^{•}(TOP/PL; ℚ) ≅ H̃^{•}(PL/O; ℚ) ≅ 0

(see Spanier [55] Ch. 9.6). The Serre spectral sequence yields thus natural isomorphisms

H^{•}(BO; ℚ) ^{≅} H^{•}(BPL; ℚ) ^{≅} H^{•}(BTOP; ℚ),

see also Kahn [31]. The rational cohomology ring of BO is known to be a polynomial algebra generated by the universal Pontrjagin classes p_{4}, p_{8}, …,

H^{•}(BO; ℚ) ≅ ℚ[p_{4}, p_{8}, p_{12}, …],

see e.g., Madsen-Milgram [41] p. 13. In view of the isomorphisms above, there are natural rational Pontrjagin classes defined for ℝ^{n}-bundles and for PL-bundles: if X BTOP(n) is a classifying map for an ℝ^{n}- bundle ξ, then p_{4k}(ξ) ∈ H^{4k}(X; ℚ) is by definition the pull-back of the universal Pontrjagin class p_{4k} ∈ H^{4k}(BTOP; ℚ) via the composite X BTOP(n) BTOP, see also Milnor-Stasheff [46] pp. 250– 251.

4.3. Recall that the signature of an oriented 4k-manifold M is by definition the signature of the quadratic form H^{2k}(M; ℚ) ℚ, v ⟼ ⟨v^{2}, [M]⟩. (The signature of a rational quadratic form represented by a symmetric matrix is the number of strictly positive eigenvalues minus the number of strictly negative real eigenvalues of the matrix.) The signature is invariant under oriented bordism and induces a group homomorphism from the oriented DIFF bordism ring into the integers,

Sig : Ω_{•}^{SO} ℤ.

It follows that the signature of a smooth closed 4k-manifold can be expressed as a certain linear combination of the rational Pontrjagin numbers of M which is given by Hirzebruch’s L-genus,

Sig(M) = ⟨L_{4k}(M), [M]⟩,

see Milnor-Stasheff [46] p. 224 and Madsen-Milgram [41] Theorem 1.38. The Hirzebruch classes L_{4k} ∈ H^{4k}(BO; ℚ) are certain rational homogeneous polynomials in the rational Pontrjagin classes, see Hirzebruch [24] 1.5, Milnor-Stasheff [46] §19. These polynomials can be obtained by a formal process from the power series expansion of the function f(t) = \begin{matrix}√ & \overline{t} \\ tanh & \sqrt{t} \end{matrix}. These results were proved by Hirzebruch for smooth closed oriented manifolds, see Hirzebruch [24] Hauptsatz 8.2.2. However, there is an isomorphism

Ω_{•}^{SO} ⊗ ℚ ≅ Ω_{•}^{STOP} ⊗ ℚ,

so the Signature Theorem carries over to the bordism ring of closed oriented manifolds, see Kahn [31], Kirby-Siebenmann [33] p. 322.

Theorem 4.4. Let M be a closed oriented 4k-manifold with fundamental class [M]. Then

Sig(M) = ⟨L_{4k}(M), [M]⟩.

The signature of our models is clearly Sig(M(ξ)) = 1.

Lemma 4.5. For our models, M(ξ), we have the following relation for the Hirzebruch classes:

L_{2m}(M(ξ)) = y_{m}^{2}.

To compute the Hirzebruch classes in terms of the Pontrjagin classes, we use the following result which is a direct consequence of Hirzebruch [24] 1.4. Let f(t) = 1 + ∑_{k≥1} f_{k}t^{k} be a formal power series, and let {K_{k}(σ_{1}, …, σ_{k})}_{k=1}^{∞} denote the corresponding multiplicative sequence, see Hirzebruch [23] §1. The associated genus F of f(t) is defined by

F_{4k} = K_{k}(p_{4}, …, p_{4k}) ∈ H^{4k}(BO; ℚ).

Hirzebruch’s L-genus comes from the formal power series

ℓ(t) = \begin{matrix}√ & \overline{t} \\ tanh & \sqrt{t} \end{matrix} = 1 + \frac{1}{3}t − \frac{1}{45}t^{2} + \frac{2}{945}t^{3} − \frac{1}{4725}t^{4} + ⋯,

and the Â-genus (which will be needed later in Section 7) from

â(t) = \frac{\sqrt{t}/2}{sinh \sqrt{t}/2}

= 1 − \frac{1}{24}t + \frac{7}{5760}t^{2} − \frac{31}{967680}t^{3} + \frac{127}{154828800}t^{4} + · · · .

Suppose now that ξ is an ℝ^{n} bundle over a space X, and that p_{4k}(ξ) and p_{8k}(ξ) are the only nonzero Pontrjagin classes of ξ. This holds for example if X is a space with H^{j}(X; ℚ) = 0 for all j ≠ 0, 4k, 8k (such as our models M(ξ)). In the cohomology ring of such a space, there is the following general formula for F_{8k} = K_{2k}(0, ⋯, 0, p_{4k}, 0, …, 0, p_{8k}). Put

f^{∨}(t) = f(t)\frac{d}{dt} (\frac{t}{f(t)}) = ∑(−1)_{k}^{ks}t^{k}.

By loc.cit. 1.4, one has the general relation

K_{2k}(0, …, 0, p_{4k}, 0, …, p_{8k}) = s_{2k}p_{8k} + \frac{1}{2}(s_{k}^{2} − s_{2k})p_{4k}^{2}

For ℓ^{∨} and â^{∨} one obtains

ℓ^{∨}(t) = 1 − \frac{1}{3}t + \frac{7}{45}t^{2} − \frac{62}{945}t^{3} + \frac{127}{4725}t^{4} + ⋯
â^{∨}(t) = 1 + \frac{1}{24}t − \frac{1}{1440}t^{2} + \frac{1}{60480}t^{3} − \frac{1}{2419200}t^{4} + · · · .

4.6. For specific values, calculations can readily be done with the formal power series package of Maple. In low dimensions, one obtains the following.

L_{4}(p_{4}) = \frac{1}{3}p_{4}
Â_{4}(p_{4}) = \frac{−1}{2^{3} · 3}p_{4}
L_{8}(p_{4}, p_{8}) = \frac{1}{3^{2} · 5}(7p_{8} − p_{4}^{2})
Â_{8}(p_{4}, p_{8}) = \frac{−1}{2^{7} · 3^{2} · 5}(4p_{8} − 7p_{4}^{2})
L_{16}(0, p_{8}, 0, p_{16}) = \frac{1}{3^{4} · 5^{2} · 7}(381p_{16} − 19p_{8}^{2})
Â_{16}(0, p_{8}, 0, p_{16}) = \frac{−1}{2^{11} · 3^{4} · 5^{2} · 7}(12p_{16} − 13p_{8}^{2}).

In the last two equations, we assume thus that p_{4} = 0 = p_{12}. See also Hirzebruch [24] 1.5 and 1.6, Milnor-Stasheff [46] p. 225, Lawson-Michelsohn [39] pp. 231–232, Eells-Kuiper [15] p. 105 for explicit formulas for these genera in low dimensions.

From the signature theorem, we obtain thus for our models strong relations between p_{m} and p_{2m}.

Lemma 4.7. For our models, we have

p_{4}(M) = \frac{1}{3}y_{2}^{2}  (m = 2)
p_{8}(M) = \frac{1}{7}(45y_{4}^{2} + p_{4}(M)^{2})  (m = 4)
p_{16}(M) = \frac{1}{381}(14175y_{8}^{2} + 19p_{8}(M)^{2})  (m = 8).

4.8. We need also the first exotic characteristic classes for TOPand PL-bundles. By a standard mapping cylinder construction, we may convert the maps BO BPL BTOP into cofibrations, see Spanier [55] Ch. 3.2 Theorem 12. Thus, it makes sense to speak of the homology and homotopy groups of the pairs (BTOP, BO) etc. Note also that these three spaces are H-spaces with isomorphic fundamental groups. Thus, π_{1}(BO) ≅ π_{1}(BPL) ≅ π_{1}(BTOP) ≅ ℤ/2 acts trivially on the homotopy groups of each of these pairs, whence π_{k} = π′_{k} for these pairs (recall that π′_{k} is the kth homotopy group, factored by the action of π_{1}, see Spanier [55] p. 390). Consequently, we have Hurewicz isomorphisms H_{k}(BPL, BO) ≅ π_{k}(BPL, BO) up to and including the lowest dimensions where the right-hand side is nontrivial, see Spanier [55] Ch. 7.5 Theorem 4. Now there is an isomorphism π_{k−1}(PL/O) ≅ π_{k}(BPL, BO), see Whitehead [66] Ch. IV 8.20. In fact, there is a commutative diagram

[figure: π_{k−1}(PL/O); ∂ ; ≅; π_{k}(C_{PL/O}, PL/O); ≅; π_{k}(BPL, BO)]

[figure: H_{k−1}(PL/O); ∂ ; ≅; H_{k}(C_{PL/O}, PL/O); H_{k}(BPL, BO)]

where C_{PL/O} ≃ ∗ is the unreduced cone. Since PL/O is 6-connected, see Madsen-Milgram p. 33, this diagram consists of isomorphisms for k ≤ 8. We obtain a similar diagram for TOP/O and TOP/PL, with isomorphisms for k ≤ 4, using the fact that TOP/PL and TOP/O are 2-connected, see Kirby-Siebenmann p. 246.

We combine this with the following result due to Hirsch [23] p. 356.

Proposition 4.9. There are short exact sequences

0  π_{k}(BO)  π_{k}(BPL)  π_{k}(BPL, BO)  0

for all k ≥ 0.

4.10. Since TOP/PL is an Eilenberg-MacLane space of type K(ℤ/2, 3), and since PL/O is 6-connected, this implies readily that each arrow

π_{k}(BO)  π_{k}(BPL)  π_{k}(BTOP)

is an injection, for all k ≥ 0.

We combine this with the commutative diagram in 4.8 and obtain thus the following result.

Lemma 4.11. There are natural isomorphisms

H_{k}(BO) ^{≅} H_{k}(BPL) ^{≅} H_{k}(BTOP)

for k ≤ 3, and H_{k}(BO) ^{≅} H_{k}(BPL) for k ≤ 7, and exact sequences

[figure: H_{4}(BO); H_{4}(BTOP); H_{4}(BTOP, BO); 0]
[figure: H_{4}(BPL); H_{4}(BTOP); H_{4}(BTOP, BPL); 0]
[figure: H_{8}(BO); H_{8}(BPL); H_{8}(BPL, BO); 0.]

Furthermore, H_{4}(BTOP, BO) ≅ H_{4}(BTOP, BPL) ≅ ℤ/2 and H_{8}(BPL, BO) ≅ ℤ/28.

Proof. The corresponding homotopy groups are given in Kirby-Siebenmann [33] p. 246. q.e.d.

From the universal coefficient theorem, see Spanier [55] Ch. 5.5 Theorem 3, we have the following result.

Proposition 4.12. For any coefficient domain R, there are exact sequences

[figure: H^{4}(BO; R); H^{4}(BTOP; R); τ_{TOP/O}; H^{3}(TOP/O; R); 0]
[figure: H^{4}(BPL; R); H^{4}(BTOP; R); τ_{TOP/PL}; H^{3}(TOP/PL; R); 0]
[figure: H^{8}(BO; R); H^{8}(BPL; R); τ_{PL/O}; H^{7}(PL/O; R); 0.]

4.13. The τ_{TOP/PL}-image of the generator of H^{3}(TOP/PL; ℤ/2) ≅ ℤ/2 is the universal Kirby-Siebenmann class ks ∈ H^{4}(BTOP; ℤ/2). For R = ℤ/4, we pick a generator κ of the τ_{PL/O}-image of H^{7}(PL/O; ℤ/4) ≅ ℤ/4. Thus we define the first exotic characteristic classes

ks ∈ H^{4}(BTOP; ℤ/2) and κ ∈ H^{8}(BPL; ℤ/4).

The map τ is the transgression (see McCleary [42] 6.2). This is maybe not obvious from our construction. To see this, consider the following diagram of cochain complexes.

0  0  0
[figure: 0; S^{•}(PL/O); S^{•}(BO); S^{•}(BO, PL/O); 0]
[figure: 0; S^{•}(C_{PL/O}); S^{•}(BPL); S^{•}(BPL, C_{PL/O}); 0]
[figure: 0; S^{•}(C_{PL/O}, PL/O); S^{•}(BPL, BO); S^{•}(BPL, BO, C_{PL/O}); 0]
[figure: 0; 0; 0]

Here, S^{•}(BPL, BO, C_{PL/O}) denotes the singular cochain complex of the triad (BPL, BO, C_{PL/O}), see Eilenberg-Steenrod [16] VII.11. Using the isomorphisms derived above, patient diagram chasing in the corresponding big diagram for cohomology (the infinite cohomology jail window, see, e.g., Cartan-Eilenberg [10] IV Proposition 2.1) shows that τ is indeed the transgression.

5. Stable ℝ^{n}-bundles over 𝕊^{m}

Our aim is the classification of ℝ^{m}-bundles over 𝕊^{m} in terms of characteristic classes. We begin with the stable classification, which is easier. Recall from 4.10 that there is an exact sequence

0  π_{k}(BO)  π_{k}(BTOP)  π_{k−1}(TOP/O)  0

for all k ≥ 0.

Lemma 5.1. In dimensions k = 2, 4, 8 these exact sequences read as follows.

[figure: 0; ℤ/2; ℤ/2; 0; 0; (k = 2); 0; ℤ; ℤ ⊕ ℤ/2; ℤ/2; 0; (k = 4); 0; ℤ; ℤ ⊕ ℤ/4; ℤ/28; 0; (k = 8).]

Proof. We consider the maps

G/TOP

[figure: TOP/O; BO; BTOP]

BG.

The space TOP/O is 2-connected, see Kirby-Siebenmann [33] p. 246, and this establishes the result for m = 2. Furthermore π_{3}(TOP/O) ≅ ℤ/2 and π_{7}(TOP/O) ≅ ℤ/28, see Kirby-Siebenmann [33] p. 246 and 200, and Kervaire-Milnor [32]. From the isomorphisms π_{k}(BG) ≅ π_{k−1}^{s}(𝕊^{0}) we have π_{4}(BG) ≅ ℤ/24 and π_{8}(BG) ≅ ℤ/240, see Toda [60] Ch. XIV, Hu [27] Ch. XI. Theorem 16.4 and p. 332, or Fomenko-Fuchs-Gutenmacher [18] p. 300. Finally, π_{4k}(BO) ≅ ℤ for k ≥ 1 by Bott periodicity. Thus we obtain diagrams

ℤ  ℤ

[figure: ℤ; mono; π_{4}(BTOP); mono; epi; ℤ/2; ℤ; mono; π_{8}(BTOP); mono; epi; ℤ/28]

[figure: epi; J_{O}; epi; J_{O}]

[figure: ℤ/24; epi; ℤ/240.; epi]

In these diagrams, the rows are short exact sequences by the remarks at the beginning of this section. The columns are also short exact, since π_{4k−1}(G/TOP) = 0, while π_{4k+1}(BG) is finite. The slanted arrows J_{O} are known to be epimorphisms in these dimensions, see e.g., Adams [1] p. 22 and p. 46.

Let T_{4k} denote the torsion group of π_{4k}(BTOP), i.e., π_{4k}(BTOP) ≅ ℤ ⊕ T_{4k}. Suppose that m = 4. The diagram shows that T_{4} injects into ℤ/2. If we tensor the diagram with ℤ/2, then J_{O} ⊗ ℤ/2 is a bijection. Therefore, the horizontal sequence

0  ℤ/2  ℤ/2 ⊕ (T_{4} ⊗ ℤ/2)  ℤ/2  0

is still exact after tensoring (the only point to check is injectivity of the second arrow). It follows that T_{4} ≅ ℤ/2.

The case m = 8 is similar. The group T_{8} injects into ℤ/28 and into ℤ/240. Since gcd(28, 240) = 4, the group injects into ℤ/4. Similarly as in the case m = 4, tensoring the diagram with ℤ/4 we see that J_{O} ⊗ℤ/4 is an isomorphism. Thus the sequence

0  ℤ/4  ℤ/4 ⊕ (T_{8} ⊗ ℤ/4)  ℤ/4  0

is exact, and T_{8} ≅ ℤ/4. q.e.d.

For k ≥ 3, the structure of the torsion groups T_{4k} was determined by Brumfiel [8], see Madsen-Milgram [41] p. 117. The cases k = 1, 2 are special; they are considered in Kirby-Siebenmann [33] p. 318 and Williamson [67] p. 29.

5.2. The result above yields thus exact sequences

[figure: 0; π_{4}(BO); ≅; π_{4}(BTOP)/T_{4}; 0; 0; (m = 4)]
[figure: 0; π_{8}(BO); π_{8}(BTOP)/T_{8}; ℤ/7; 0; (m = 8).]

The cokernels of the corresponding maps π_{4k}(BO) π_{4k}(BTOP)/T_{4k} for k ≥ 3 are determined in Brumfiel [8] p. 304 in number theoretic terms.

Consider now the map which assigns to a stable ℝ^{n}-bundle ξ over 𝕊^{4k} the rational Pontrjagin number ⟨p_{4k}(ξ), [𝕊^{4k}]⟩. We want to determine the possible values of this map, and we do this first for vector bundles.

5.3. For a finite connected CW-complex X and an n-dimensional vector bundle ξ over X, there is a classifying map X BO(n). The Pontrjagin classes of X are obtained by pulling back the universal Pontrjagin classes in H^{•}(BO; ℚ) via the composite X BO(n) BO. Only the homotopy type of this map is important, so we view it as an element of the set [X; BO] of free homotopy classes of maps from X to BO (since BO is an H-space and X is connected, we have [X; BO]_{0} = [X, BO], as discussed in the remarks at the end of the introduction of this paper). Now this set [X; BO] can be identified with the real reduced K̃O-theory of X,

[X; BO] = K̃O(X),

see Atiyah-Hirzebruch [3], Hirzebruch [25] or Husemoller [30]. The Pontrjagin character is the ring homomorphism ph = ch ∘ cplx,

KO(X) ^{cplx} KU(X) ^{ch} H^{•}(X; ℚ)

where cplx denotes complexification of real vector bundles, and ch denotes the Chern character of complex KU-theory, see Atiyah-Hirzebruch [3] or Hirzebruch [25] 1.4. For the following facts see Hirzebruch [25] 1.4–1.6. Hirzebruch’s integrality theorem says that

ch(KU(𝕊^{2k})) = H^{•}(𝕊^{2k}) ⊆ H^{•}(𝕊^{2k}; ℚ),

with ch(η) = rk_{ℂ}(η) + (−1)^{k−1} \frac{1}{(k−1)!}c_{k}(η). Recall also that p_{4k}(ξ) = (−1)^{k}c_{2k}(ξ ⊗ ℂ). Thus we have the formula

ph(ξ) = rk_{ℝ}(ξ) + (−1)^{k−1} \frac{1}{(2k − 1)!}p_{4k}(ξ)

on 𝕊^{4k}. The map K̃O(𝕊^{4k}) ^{cplx} K̃U(𝕊^{4k}) is injective, with cokernel 0 for k even, and cokernel ℤ/2 for k odd.

Combining these facts, we have the following result.

Lemma 5.4. Let ξ be a vector bundle over 𝕊^{4k}, and let x ∈ H^{4k}(𝕊^{4k}) be a generator. Then

p_{4k}(ξ) = a_{ξ} · d_{k} · (2k − 1)! · x

where a_{ξ} is an integer depending on ξ, and d_{k} = 1 for k even, d_{k} = 2 for k odd. Conversely, given any integer a_{ξ}, there exists a vector bundle ξ with such a Pontrjagin class, and ξ is unique up to stable equivalence.

Proof. Since BO is an H-space, the set [𝕊^{4k}; BO] of free homotopy classes coincides with the homotopy group π_{4k}(BO) = [𝕊^{4k}; BO]_{0}. Viewing the universal Pontrjagin class p_{4k} as a map BTOP ^{p_{4k}} K(ℚ, 4k) into an Eilenberg-MacLane space, we obtain a nontrivial group homomorphism

ℤ ≅ π_{4k}(BO) ^{(p_{4k})_{#}} π_{4k}(K(ℚ, 4k)) ≅ ℚ.

q.e.d.

If ξ is a vector bundle, the possible values of the rational numbers ⟨p_{4k}(ξ), [𝕊^{m}]⟩ are thus the integral multiples of d_{k} · (2k − 1)!.

Lemma 5.5. Let ξ be an ℝ^{n}-bundle over 𝕊^{m}, for m = 4, 8. Then

⟨p_{4}(ξ), [𝕊^{4}]⟩ ∈ 2ℤ ⊆ ℚ
⟨p_{8}(ξ), [𝕊^{8}]⟩ ∈ \frac{6}{7}ℤ ⊆ ℚ.

Conversely, for each of these values, there exists an ℝ^{n}-bundle (for some sufficiently large n) whose Pontrjagin number assumes this value.

Proof. This is clear from Lemma 5.4, applied to the special cases k = 1, 2, and the formula for the cokernel of the map

π_{4k}(BO) π_{4k}(BTOP)/T_{4k}

which was derived in 5.2. q.e.d.

We have proved the following result.

Proposition 5.6. Let ξ be an ℝ^{n}-bundle over 𝕊^{m}, for m = 2, 4, 8. Up to stable equivalence, the bundle ξ is completely determined by the following characteristic classes:

(m = 2) its 2nd Stiefel-Whitney class w_{2}(ξ).

(m = 4) its 4-dimensional Pontrjagin class p_{4}(ξ) and its Kirby-Sieben-mann class ks(ξ).

(m = 8) its 8-dimensional Pontrjagin class p_{8}(ξ) and the characteristic class κ(ξ).

The possible ranges for the values of these characteristic classes, evaluated on the fundamental class [𝕊^{m}], are ℤ/2 (for m = 2), 2ℤ and ℤ/2 (for m = 4), and \frac{6}{7}ℤ and ℤ/4 (for m = 4), respectively.

Proof. We prove the 8-dimensional case; the others are similar. Let (BPL, BO) (K(ℤ/4, 8), ∗) represent the generator of H^{8}(BPL, BO; ℤ/4) which maps to κ. Then the composite BPL (BPL, BO) (K(ℤ/4, 8), ∗) induces an isomorphisms on the ℤ/4-factors in

π_{8}(BPL)  π_{8}(BPL, BO)  π_{8}(K(ℤ/4, 8)),

which can be identified with the map ξ ⟼ κ(ξ) ∈ H^{8}(BPL; ℤ/4). The octonionic Hopf bundle η_{𝕆} over 𝕊^{8} represents an element of π_{8}(BPL) with p_{8}(η_{𝕆}) ≠ 0. Thus, the map ξ ↦ (p_{8}(ξ), κ(ξ)) ∈ H^{8}(𝕊^{8}; ℚ) ⊕ H^{8}(𝕊^{8}; ℤ/4) ≅ ℚ ⊕ ℤ/4 is an injection (with image \frac{6}{7}ℤ ⊕ ℤ/4). q.e.d.

6. ℝ^{m}-bundles over 𝕊^{m}

In the previous section, we classified bundles over 𝕊^{m} up to stable equivalence in terms of characteristic classes. To obtain an unstable classification, i.e., a classification of ℝ^{m}-bundles over 𝕊^{m}, we use the following result.

Proposition 6.1. Let m ≥ 2 be even. Then there is a commutative diagram with exact rows

[figure: 0; π_{m}(𝕊^{m}); π_{m}(BO(m)); π_{m}(BO); 0]

[figure: 0; π_{m}(𝕊^{m}); ‖; π_{m}(BPL(m)); π_{m}(BPL); 0]

[figure: 0; π_{m}(𝕊^{m}); ‖; π_{m}(BTOP(m)); π_{m}(BTOP); 0]

[figure: 0; π_{m}(𝕊^{m}); ‖; π_{m}(BG(m)); π_{m}(BG); 0.]

In this diagram, the second column of vertical arrows is induced by the respective classifying map of the tangent bundle of 𝕊^{m} (resp. its underlying spherical fibration) and the third column of vertical arrows corresponds to stabilization.

Proof. This is proved in [37] pp. 93–95; there, the result is stated for the oriented case, but the homotopy groups are the same. The PL result is not stated in [37], but in low dimensions, π_{k}(BO) ≅ π_{k}(BPL) and in higher dimensions π_{k}(BPL) ≅ π_{k}(BTOP), see Kirby-Siebenmann [33] V §5. For a related result (but excluding dimension 4) see Varadarajan [61]. q.e.d.

Our aim is to prove that the first three rows in this diagram split, provided that m ≥ 4. It suffices to prove this for the third row; the diagram then implies the splitting of the first two rows.

Lemma 6.2. Let m ≥ 4 be even. Then the exact sequence

0  π_{m}(𝕊^{m})  π_{m}(BTOP(m))  π_{m}(BTOP)  0

splits.

Proof. Since we consider only higher dimensional homotopy groups, we may as well consider the universal coverings BSTOP(m) and BSTOP which classify oriented bundles. So we have an exact sequence

0  π_{m}(𝕊^{m})  π_{m}(BSTOP(m))  π_{m}(BSTOP)  0.

Let e denote the universal Euler class, viewed as a map BSTOP(m) ^{e} K(ℤ, m) into an Eilenberg-MacLane space. The Euler class yields thus a homomorphism

π_{m}(BSTOP(m)) ^{e_{#}} π_{m}(K(ℤ, m)) = H^{m}(𝕊^{m}) ⟨−,[𝕊^{m}]⟩ ; ≅ ℤ.

If m ≠ 4, 8, then the image of this map is 2ℤ by Adams’ result (see Proposition 2.3). The Euler class of the tangent bundle of 𝕊^{m} is 2x, so the exact sequence above splits: we have constructed a left inverse for the second arrow π_{m}(𝕊^{m}) π_{m}(BSTOP(m)).

For m = 4, 8, we consider the last two rows of the big diagram,

[figure: 0; π_{m}(𝕊^{m}); π_{m}(BTOP(m)); π_{m}(BTOP); 0]

[figure: 0; π_{m}(𝕊^{m}); ‖; π_{m}(BG(m)); π_{m}(BG); 0.]

Let T′_{m} denote the torsion subgroup in π_{m}(BTOP(m)), and T_{m} ≅ ℤ/(m/2) the torsion subgroup of π_{m}(BTOP)). It is clear that T′_{m} injects into T_{m}. The sequence splits if and only if T′_{m} maps isomorphically onto T_{m}. Now π_{4}(BG(4)) ≅ π_{3}(G(4)) ≅ π_{7}(𝕊^{4}) ≅ ℤ ⊕ ℤ/12, and π_{4}(BG) ≅ π_{3}^{s}(𝕊^{0}) ≅ ℤ/24. Similarly, π_{8}(BG(8)) ≅ π_{7}(G(8)) ≅ π_{15}(𝕊^{8}) ≅ ℤ ⊕ ℤ/120, and π_{8}(BG) ≅ π_{7}^{s}(𝕊^{0}) ≅ ℤ/240. See Toda [60] Ch. XIV, Hu [27] Ch. XI. Theorem 16.4 and p. 332, or Fomenko-Fuchs-Gutenmacher [18] p. 300 for these groups. It follows that in the bottom row of the diagram, a generator ι_{m} ∈ π_{m}(𝕊^{m}) maps to (2, −1) ∈ ℤ⊕ℤ/12 resp. in ℤ ⊕ ℤ/120, for m = 4, 8. If we tensor the diagram above with ℤ/2 for m = 4 (resp. with ℤ/4 for m = 8), then the image of ι_{m} still has order 2 (resp. 4) in π_{m}(BG(m)) ⊗ ℤ/(m/2). Thus the bottom row remains exact after tensoring, and therefore, the upper row remains also exact (the only point to be checked was the injectivity of the second arrow). It follows that T′_{m} ≅ T_{m}. q.e.d.

In dimension m = 2, we use Kneser’s old result BO(2) ≃ BTOP(2), see Kirby-Siebenmann [33] p. 254. Thus, there is a (non-split) short exact sequence

0  π_{2}(𝕊^{2})  π_{2}(BTOP(2))  ℤ/2  0

and π_{2}(BTOP(2)) ≅ ℤ. Using similar ideas as above, it is not difficult to prove that the sequence

0  π_{m}(𝕊^{m})  π_{m}(BG(m))  π_{m}(BG)  0

splits for all even m ≠ 2, 4, 8 (and for these three values, the sequence is not split). We will not need this result.

6.3. Proof of Proposition 3.3. Let ξ and ξ′ be ℝ^{m}-bundles over 𝕊^{m}, for m ≥ 2 even. Assume first that we can choose orientations of these bundles such that e(ξ) = e(ξ′). Let c denote the classifying map for the oriented tangent bundle τ𝕊^{m}, and let c_{ξ} and c_{ξ′} be classifying maps for the oriented bundles ξ and ξ′. The splitting of the exact sequence

0  π_{m}(𝕊^{m}) ^{c_{#}} π_{m}(BSTOP(m))  π_{m}(BSTOP)  0

implies then that (c_{ξ})_{#} = (c_{ξ′})_{#} (as maps π_{m}(𝕊^{m}) π_{m}(BSTOP)). Thus ξ ≅ ξ′.

In the general case m ≥ 4, we have the action of π_{1}(BO(m)) ≅ π_{1}(BTOP(m)) ≅ ℤ/2 on the higher homotopy groups. The generator α_{0} of the fundamental group of BO(m) maps c_{#} to its negative −c_{#} (see Steenrod [57] 23.11). From the splitting of the exact sequence

0  π_{m}(𝕊^{m})  π_{m}(BTOP(m))  π_{m}(BTOP)  0

and the diagram in Proposition 6.1 we see that α_{0} changes the sign of the Euler class. Thus, if |e| = |e′|, then we may as well assume that e(ξ) = e(ξ′).

The case m = 2 follows directly from BO(2) ≃ BTOP(2). q.e.d.

We summarize our classification of ℝ^{m}-bundles over 𝕊^{m}, for m = 2, 4, 8, as follows.

Proposition 6.4. Let ξ be an ℝ^{2}-bundle over 𝕊^{2}. Up to equivalence, ξ is determined by its absolute Euler number |e| = |⟨e(ξ), [𝕊^{2}]⟩|, and for each |e| ∈ ℕ, there exists one such bundle. A weak equivalence between any two such bundles is an equivalence.

Let ξ be an ℝ^{4}-bundle over 𝕊^{4}. Up to equivalence, ξ is determined by its absolute Euler number |e| = |⟨e(ξ), [𝕊^{2}]⟩|, its Kirby-Siebenmann number ⟨ks(ξ), [𝕊^{4}]⟩ ∈ ℤ/2 and the Pontrjagin number ⟨p_{4}(ξ), [𝕊^{4}]⟩ ∈ 2 · ℤ. For each triple (|e|, ks, p_{4}) ∈ ℕ × ℤ/2 × 2 · ℤ satisfying the relation p_{4} + 2|e| ≡ 0 (mod 4), there exists one such bundle. If two such bundles ξ, ξ′ are weakly equivalent, but not equivalent, then (|e|, p_{4}, ks) = (|e′|, −p′_{4}, ks′).

Let ξ be an ℝ^{8}-bundle over 𝕊^{8}. Up to equivalence, ξ is determined by its absolute Euler number |e| = |⟨e(ξ), [𝕊^{2}]⟩|, the number ⟨κ(ξ), [𝕊^{8}]⟩ ∈ ℤ/4 and the Pontrjagin number ⟨p_{8}(ξ), [𝕊^{8}]⟩ ∈ \frac{6}{7} · ℤ. For each triple in ℕ × ℤ/4 × \frac{6}{7} · ℤ satisfying the relation \frac{7}{3}p_{8} + 2|e| ≡ 0 (mod 4), there exists one such bundle. If two such bundles ξ, ξ′ are weakly equivalent, but not equivalent, then (|e|, p_{8}, κ) = (|e′|, −p′_{8}, −κ).

Proof. We prove the 8-dimensional case; the other cases are similar. First, we classify oriented bundles; the orientation we choose is the orientation determined by the universal oriented ℝ^{8}-bundle over BSTOP(8). Then it is clear from our discussion that ξ is determined by the data

(⟨e(ξ), [𝕊^{8}]⟩, ⟨κ(ξ), [𝕊^{8}]⟩, ⟨p_{8}(ξ), [𝕊^{8}]⟩) ∈ ℤ × ℤ/4 × \frac{6}{7}ℤ.

Now we have as in 6.3 the action of α_{0} which changes the sign of the Euler class without changing the sign of p_{8} and κ (since these two classes come from BTOP, where α_{0} acts trivially). This shows that the given numbers classify the bundle up to equivalence.

Let ι_{8} ∈ π_{8}(𝕊^{8}) denote the canonical generator, and let c be the classifying map for the oriented tangent bundle of 𝕊^{8}. For the numbertheoretic relation between the Pontrjagin class and the Euler class, we note first that the image (c)_{#}(π_{8}(𝕊^{8})) is a direct factor in π_{8}(BSTOP(8)). The octonionic Hopf line bundle η_{𝕆} has Euler class x and Pontrjagin class 6x (for a suitable generator x of H^{m}(𝕊^{m})), see, e.g., [36] Theorem 9. Let h be a classifying map for the oriented bundle η_{𝕆}. Then c_{#}(ι_{8}) and h_{#}(ι_{8}) span π_{8}(BSTOP(8)) ⊗ ℚ (over ℚ), and the image of h in π_{8}(BSTOP) spans π_{8}(BSTOP) ⊗ ℚ (over ℚ). Since e(ξ) is necessarily integral for any oriented ℝ^{8}-bundle over 𝕊^{8}, we see from Lemma 5.5 that there exists a bundle η′ with classifying map h′ and e(η′) = x, p_{8}(η′) = \frac{6}{7}x, and that c_{#}(ι_{8}) and h′_{#}(ι_{8}) span a direct complement of the torsion group of π_{8}(BSTOP(8)) (over ℤ).

Finally, a weak equivalence which is not an equivalence comes from a map 𝕊^{8} 𝕊^{8} of degree −1; such a map changes the sign of every characteristic class. q.e.d.

7. The classification of the models

In this section, we obtain the final homeomorphism classification of our models. We fix some notation. Let ξ be an ℝ^{m}-bundle over 𝕊^{m}, with absolute Euler number |e| = 1, for m = 2, 4, 8, let M(ξ) be its Thom space, and let s_{0} : 𝕊^{m} E ⊆ M(ξ) be the zero-section. Let y_{m} ∈ H^{m}(M(ξ)) denote a generator, such that x = s_{0}^{•}y_{m} is a generator dual to the chosen orientation [𝕊^{m}]. For m = 4, 8, we have by 3.7 the relations

s_{0}^{•}(p_{m}(M(ξ))) = p_{m}(ξ)  (m = 4, 8)
s_{0}^{•}(ks(M(ξ))) = ks(ξ)  (m = 4)
s_{0}^{•}(κ(M(ξ))) = κ(ξ)  (m = 8).

Theorem 7.1. Up to homeomorphism, our construction yields precisely the following models.

For m = 2, there is just one model, the complex projective plane M(η_{ℂ}) ≅ ℂP^{2}, where η_{ℂ} is the complex Hopf line bundle (the tautological bundle) over ℂP^{1} = 𝕊^{2}.

For m = 4, let p_{4}(ξ) = 2(1 + 2t)x and ks(ξ) = sx, for (t, s) ∈ ℤ × ℤ/2. By Proposition 6.4, the pair (r, s) = (1 + 2t, s) determines ξ up to equivalence, so we may put M_{r,s} = M(ξ). If there is a homeomorphism M_{r,s} ≅ M_{r′,s′}, then (r, s) = (±r′, s). The quaternion projective plane is M_{1,0} ≅ ℍP^{2} = M(η_{ℍ}) ≅ M_{−1,0}. The model M_{r,s} admits a PL structure (unique up to isotopy) if and only if s = 0.

For m = 8, let p_{8}(ξ) = \frac{6}{7}(1+2t)x and κ(ξ) = sx, for (t, s) ∈ ℤ×ℤ/4. Again, the pair (r, s) = (1 + 2t, s) determines ξ up to equivalence by Proposition 6.4, so we may put M_{r,s} = M(ξ). If there is a homeomorphism M_{r,s} ≅ M_{r′,s′}, then (r, s) = ±(r′, s′). The octonionic projective plane is M_{7,0} ≅ 𝕆P^{2} = M(η_{𝕆}) ≅ M_{−7,0}. Each model M_{r,s} admits a PL structure (unique up to isotopy).

Proof. The topological result is clear from our classification of ℝ^{m}- bundles over 𝕊^{m} with absolute Euler number |e| = 1. For a closed manifold M of dimension at least 5, the only obstruction to the existence of a PL structure is the Kirby-Siebenmann class ks(M). If such a PL structure exists on M , the number of isotopy classes of PL structures is determined by [M; TOP/PL] ≅ H^{3}(M; ℤ/2), see Kirby-Siebenmann [33] p. 318. In our case, these groups are zero. q.e.d.

The models constructed by Eells-Kuiper in [14] are ℂP^{2} (for m = 2), the models M_{r,0}, with r = 1 + 2t (for m = 4), and the models M_{7r,0}, with r = 1 + 2t (for 8 = 4). Brehm-Kühnel [6] construct 8-dimensional PL manifolds which look like projective planes, and with small numbers of vertices. However, the characteristic classes of their examples seem to be unknown.

7.2. Now we consider the question which of the models admit a DIFF structure. The number of PL or DIFF structures on ℂP^{2} is presently not known, so we concentrate from now on the cases m = 4, 8. Concerning DIFF structures in higher dimensions, a necessary condition (besides the existence of a PL structure) is clearly that τM(ξ) admits a vector bundle structure. In particular, ξ ≅ s_{0}^{∗}τM(ξ) has to admit a vector bundle structure. Thus, if M_{r,s} = M_{1+2t,s} admits a DIFF structure, then clearly s = 0, and in addition t ≡ 3 (mod 7) for m = 8. But this guarantees only that ξ admits a vector bundle structure. From 4.6 and Lemma 4.7, we see that

Â [M_{1+2t,s}] = −\frac{t(1 + t)}{2^{3} · 7}  (m = 4)
Â [M_{7(1+2u),s}] = −\frac{u(1 + u)}{2^{7} · 127}  (m = 8).

Since M_{r,s} is 3-connected, the first Stiefel-Whitney classes vanish. Thus, if M_{r,s} admits a DIFF structure, then it is a Spin manifold, see Lawson-Michelsohn [39] Ch. II Theorem 2.1. But for a closed oriented Spin manifold M^{4k}, the Â-genus Â[M] is precisely the index of the Atiyah-Singer operator, see Lawson-Michelsohn [39] Ch. IV Theorem 1.1; in particular, it is an integer.

Lemma 7.3. If M_{r,s} admits a DIFF structure, then we have the following relations: For m = 4 put r = 1 + 2t. Then s = 0 and t ≡ 0, 7, 48, 55 (mod 56). For m = 8 put r = 7(1 + 2u). Then s = 0 and u is an integer with u ≡ 0, 127, 16128, 16255 (mod 16256).

This corresponds to Eells-Kuiper [14] Proposition 10 A, p. 43. In fact, the above conditions are sharp.

Theorem 7.4. If m = 4, then M_{1+2t,s} admits a DIFF structure if and only if s = 0 and t ≡ 0, 7, 48, 55 (mod 56). If m = 8, then M_{7(1+2u),s} admits a DIFF structure if and only if s = 0 and u ≡ 0, 127, 16128, 16255 (mod 16256).

Proof. The number theoretic condition guarantees that ξ admits a vector bundle structure. Thus we may choose a Riemannian metric on ξ. Let SE 𝕊^{m} denote the corresponding unit sphere bundle of ξ. Then E_{0} ≃ SE is a homotopy 2m − 1-sphere and thus homeomorphic to 𝕊^{2m−1} by the proof of the generalized Poincaré conjecture, see Smale [54] and Newman [47]. If SE is diffeomorphic to 𝕊^{2m−1}, then we may choose a diffeomorphism α : 𝕊^{2m−1} SE. Gluing the closed 2m-disk 𝔻^{2m} along α to the closed unit disk bundle DE of the vector bundle ξ, we obtain a smooth 2m-manifold DE ∪_{α} 𝔻^{2m} homeomorphic to M_{r,s} ≅ DE/SE. Thus the problem is reduced to the question whether SE is diffeomorphic to 𝕊^{2m}. Now the unit disk bundle X = DE is an almost closed manifold, i.e., a smooth compact manifold X with boundary ∂X = SE a homotopy sphere, see Wall [62]. By loc.cit. p. 178, such a manifold X^{2m} has a standard sphere 𝕊^{2m−1} as its boundary if and only if its Â-genus is integral, provided that m = 4, 8 and that X is m − 1-connected. The Â-genus of DE coincides of course with the Â-genus of M_{r,s}. q.e.d.

The case m = 8 remained open in Eells-Kuiper [14]. Note that the existence of a positive scalar curvature metric implies that the Â-genus vanishes, see Lawson-Michelsohn [39] Ch. IV Theorem 4.1; this happens only for the models ℍP^{2} and 𝕆P^{2}.

Proposition 7.5. The only models which admit a DIFF structure with a positive scalar curvature metric are ℍP^{2} and 𝕆P^{2}.

These two manifolds admit a positive scalar curvature metric for any DIFF structure; this follows from Stolz’ proof of the Gromov-Lawson-Rosenberg conjecture, see Stolz [59] Theorem A (for these two manifolds, Rosenberg’s earlier result [51] for lower dimensions actually suffices). For spin manifolds of dimension 4k, the map α : Ω_{4k}^{Spin} KO(𝕊^{4k}) can be identified with a scalar multiple of the Â-genus.

Concerning the number of DIFF structures on a smoothable model M_{r,s}, we have Wall’s result [62] which says that the DIFF structure on the almost closed manifold DE is unique. The group Θ_{2m} then acts transitively on the collection of all smoothings of M_{r,s}, see the introduction in Stolz [58]. This group is cyclic of order 2 for m = 4, 8, see Kervaire-Milnor [32] p. 504.

Proposition 7.6. If a model M_{r,s}, for m = 4, 8, admits a DIFF structure, then it admits at most two distinct DIFF structures.

This fact was already observed in Eells-Kuiper [14]. To obtain a more precise result, i.e., the exact number of DIFF structures, one would have to determine the inertia groups of the smoothable models (see Stolz [58]). Maybe his high-dimensional techniques can be adapted to this situation.

7.7. Finally, we consider oriented bordisms between distinct models. So suppose that

∂W = M_{r,s} ∪ −M_{r′,s′}

is a compact oriented (topological) bordism between two models. Clearly, the numbers p_{m}^{2}[M_{r,s}], ks^{2}[M_{r,s}] (for m = 4), and κ^{2}[M_{r,s}] (for m = 8) are bordism invariants. Thus, the existence of an oriented bordism implies that (r^{2}, s^{2}) = (r′^{2}, s′^{2}). This is good enough to settle the case m = 4; here, we conclude that (r, s) = ±(r′, s′) and thus M_{r,s} ≅ M_{r′,s′}, because s ∈ ℤ/2 has no sign. For m = 8 this is not good enough to conclude that (r, s) = ±(r′, s′) because s is ℤ/4-valued. So we use the standard fact from topological surgery theory (as developed in Kirby-Siebenmann [33]) that such a bordism W can be made 7connected. The classifying map W BSTOP for the oriented tangent bundle lifts thus to the 7-connected cover BSTOP⟨8⟩. Put π = π_{8}(BSTOP) ≅ ℤ ⊕ ℤ/4 and let BSTOP⟨8⟩ K(π, 8) denote the corresponding characteristic map. Let x ∈ H^{8}(K(π, 8)) ≅ ℤ ⊕ ℤ/4 be a generator for a free cyclic factor, and let q_{8} denote its image in H^{8}(BSTOP⟨8⟩). If X is any 7-connected CW-complex, and if ξ is a stable ℝ^{n}-bundle over X, then the classifying map X ^{c} BSTOP lifts

BSTOP⟨8⟩ \overline{q_{8}} K(π, 8)

....................

[figure: X; c; BSTOP]

and the class q_{8}(ξ) is defined. From the coefficient pairing ℤ⊗ℤ/4 ℤ/4, we obtain for any ℝ^{n}-bundle over a 7-connected space X a ℤ/4valued 16-dimensional characteristic class q_{8}κ and clearly, this class is a bordism invariant for 7-connected oriented bordisms. For X = 𝕊^{8}, we know that q_{8}(ξ) = \frac{7}{6}p_{8}(ξ). For our 16-dimensional models, we have thus q_{8}(M_{r,s}) = \frac{7}{6}p_{8}(M) by 3.7, and this is an odd integral multiple of the generator y_{8} ∈ H^{8}(M_{r,s}). The 7-connected bordism yields now the additional relation \frac{7}{6}p_{8}κ[M_{r,s}] = \frac{7}{6}p_{8}κ[M_{r′,s′}], which implies that (r, s) = ±(r′, s′).

Proposition 7.8. Non-homeomorphic models M_{r,s}, M_{r′,s′} fall into different oriented bordism classes in Ω_{2m}^{STOP}.

8. The homotopy types of the models

For the homotopy classification of our models we use the Spivak fibration. We recall the construction and refer to Klein [34] for more details. Let M be a closed 1-connected manifold (the case where π_{1}(M) ≠ 1 is more involved and will not be important to us). There exists an embedding M ^{⊂} 𝕊^{N}, for some sufficiently large N, such that M hat a normal bundle νM in 𝕊^{N}. In the group { \atop KTOP}(M) = [M; BTOP], the bundle νM is just the inverse of the stable bundle class determined by the tangent bundle τM, since τM ⊕ νM ≅ τ𝕊^{N}|_{M} = \underline{ℝ}^{N}. This shows that the normal bundle νM is unique up to stable equivalence.

There is a natural map 𝕊^{N} ^{α} M(νM), where M(νM) is the Thom space of the normal bundle. Let u(νM) be an orientation class. One shows that for the fundamental classes of 𝕊^{N} and M, one has the relation

u(νM) ⌢ α_{•}[𝕊^{N}] = [M]

(for the right choice of u(νM)). Now a result by Spivak [56] shows that the stable fiber homotopy type of the underlying spherical fibration σM of the bundle νM depends only on the homotopy type of M.

8.1. Let c : M BTOP be a stable classifying map for νM, and let d : M BTOP be a stable classifying map for τM. Then c is an inverse of d in the abelian group { \atop KTOP}(M) = [M; BTOP], and so the composites M ^{c} BTOP BG and M ^{d} BTOP BG are inverse to each other in the abelian group [M; BG]. But M ^{c} BTOP

BG is a classifying map for σM and depends thus by Spivak’s result only on the homotopy type of M. It follows that the composite M ^{d} BTOP BG is also a homotopy invariant of M.

Proposition 8.2. If there is a homotopy equivalence f : M(ξ) ^{≃} M(ξ′) between two models, then there is a fiber homotopy equivalence τM(ξ) ⊕ \underline{ℝ}^{k} ≃ f^{∗}τM(ξ′) ⊕ \underline{ℝ}^{k}, for some k ≥ 0.

8.3. Since s_{0} : 𝕊^{m} M(ξ) represents a generator of π_{m}(M(ξ)), this has the consequence that there is a fiber homotopy equivalence

ξ ⊕ \underline{ℝ}^{k} ≃ g^{∗}ξ′ ⊕ \underline{ℝ}^{k},

for some homeomorphism g : 𝕊^{m} 𝕊^{m} of degree ±1. Since ξ has absolute Euler number |e| = 1, the mth Stiefel-Whitney class of ξ is nontrivial: the mth Stiefel-Whitney class is the mod 2 reduction of the Euler class, see Milnor-Stasheff [46] Proposition 9.5, so 0 ≠ w_{m}(ξ) = x ∈ H^{m}(𝕊; ℤ/2). Also, the Stiefel-Whitney class depends only on the stable type of the spherical fibration of ξ. Let R_{m} ⊆ π_{m}(BG) denote the set of all elements which represent a spherical fibration over 𝕊^{m} with nontrivial mth Stiefel-Whitney class. This is a coset of a subgroup of index 2 in π_{m}(BG) (namely the kernel of the map π_{m}(BG) ^{(w_{m})_{#}} π_{m}(K(m, ℤ/2))). Precomposing the classifying map with a map of degree −1, we achieve a change of sign for all elements in π_{m}(BG). The group π_{m}(BG) ≅ π_{m−1}^{s}(𝕊^{0}) is cyclic of order 2, 24, 240, for m = 2, 4, 8, respectively, see Toda [60] Ch. XIV, Hu [27] Ch. XI. Theorem 16.4 and p. 332, or Fomenko-Fuchs-Gutenmacher [18] p. 300. Thus we see that there are at least 1, 6, 60 distinct homotopy types which are realized by our models, for m = 2, 4, 8. In the appendix we prove that these numbers are the precise numbers of homotopy types of Poincaré duality complexes (see 9.3) which look like projective planes.

Theorem 8.4. Every 1-connected Poincaré duality complex which looks like a projective plane is homotopy equivalent to one of our models. The homotopy type of a model M_{r,s} can be determined as follows. For m = 2, there is just one model and one homotopy type, namely ℂP^{2}.

If m = 4, then M_{r,s} ≃ M_{r′,s′} if and only if r + 12s ≡ ±(r′ + 12s′) (mod 24).

If m = 8, then M_{r,s} ≃ M_{r′,s′} if and only if r + 60s ≡ ±(r′ + 60s′) (mod 240).

9. Our set of models is complete

In this section we prove that every manifold which looks like a projective plane is homeomorphic to one of our model manifolds M(ξ) — except for dimension 4, where one has precisely two such manifolds, the model manifold ℂP^{2} = M(η_{ℂ}) and in addition the Chern manifold Ch^{4} which is not a Thom space, see Theorem 9.1. This is covered by Freedman’s classification [19] of closed 1-connected 4-manifolds.

Theorem 9.1. There are precisely two 1-connected closed 4-mani-folds M with H_{•}(M) ≅ ℤ^{3}, the complex projective plane ℂP^{2} and the Chern manifold Ch^{4}.

Proof. This is stated in Freedman [19] pp. 370–372. Such a manifold is represented by the odd integral symmetric bilinear form ω = (1) on H_{2}(M) ≅ ℤ and its Kirby-Siebenmann number ks[M] ∈ ℤ/2. q.e.d.

From now on, we assume that m ≠ 2. The main result of this section is the following.

Theorem 9.2. Let M^{2m} be a manifold which is like a projective plane. If m ≠ 2, then M is homeomorphic to one of our models M(ξ).

The proof requires surgery techniques, so we recall the relevant notions. More information can be found in Madsen-Milgram [41] Ch. 2, in Kirby-Siebenmann [33] Essay V App. B, in Wall [64] Ch. 10, and in particular in Kreck [38]. The basic fact to keep in mind is that by the results of Kirby-Siebenmann [33], higher dimensional surgery works well in the topological category. The case m = 2 can in principle be handled by similar methods, see Freedman-Quinn [20].

9.3. The spaces we are dealing with are 1-connected, and this simplifies some points. Suppose that X is a finite and 1-connected CWcomplex. Assume moreover that there is an element [X] ∈ H_{n}(X) such that the cap product induces an isomorphism

H^{k}(X) ⌢[X] ; ≅ H_{n−k}(X)

for all k. Then X satisfies Poincaré duality, and the pair (X, [X]) is what is called a Poincaré duality complex (of formal dimension n). Every closed, 1-connected and oriented manifold is a Poincaré duality complex (we consider here only the 1-connected case; the presence of a fundamental group requires the more complicated notion of a simple homotopy type).

9.4. Next, recall that an h-cobordism (W; M_{1}, M_{2}) is a simply connected compact bordism between (simply connected) closed manifolds M_{1}, M_{2},

∂W = M_{1} ˙∪M_{2},

with the property that the inclusions M_{1}  W  M_{2} are homotopy

equivalences; an example is the product cobordism, (M × [0, 1], M × 0, M × 1). In higher dimensions, this is in fact the only example.

9.5. h-cobordism Theorem. Every h-cobordism (W; M_{1}, M_{2}) with dim(W) ≥ 5 is a product bordism, W ≅ M_{1} × [0, 1]. In particular, there is a homeomorphism M_{1} ≅ M_{2}.

Proof. For dim W ≥ 6, this is proved in Kirby-Siebenmann [33], but unfortunately not stated explicitly as a Theorem; see loc. cit. p. 113 and p. 320. For dim(W) ≥ 7, a proof is given by Okabe [48]. The case dim(W) = 5 is proved in Freedman-Quinn [20], with some remarks on the higher dimensional case. q.e.d.

9.6. Suppose now that M is a closed oriented manifold of dimension at least 5, and that f : M X is a homotopy equivalence, with f_{•}[M] = [X]. Then f is called a homotopy manifold structure on X; two such homotopy manifold structures M_{1} ^{f_{1}} X ^{f_{2}} M_{2} are called equivalent if there exists an h-cobordism (W; M_{1}, M_{2}) and a map F : W X such that the diagram

[figure: M_{1}; ⊂; W; ⊃; M_{2}]
[figure: f_{1}; f_{2}]
X \begin{aligned}F \\ &  \end{aligned}

commutes. This relation is transitive and symmetric; the set of all equivalence classes of homotopy manifold structures on X is the structure set S_{TOP}(X). Since we are assuming that dim(M) ≥ 5, the h-cobordism Theorem 9.5 applies, and thus every element of S_{TOP}(X) represents a well-defined homeomorphism type of a closed manifold homotopy equivalent to X.

Let Aut(X) ⊆ [X; X] denote the group of all self-equivalences of X. If S_{TOP}(X) is nonempty, there is a natural action of Aut(X) on S_{TOP}(X), and the orbit set

M_{TOP}(X) = S_{TOP}(X)/Aut(X)

can be identified with the set of all homeomorphism types of manifolds homotopy equivalent with X.

9.7. In order to determine the structure set S_{TOP}(X), it is convenient to introduce yet another set which contains S_{TOP}(X) as a subset, the set T_{TOP}(X) of tangential invariants. Recall from Section 8 that associated to a 1-connected Poincaré duality complex X is a spherical fibration, the Spivak normal bundle σX whose stable fiber homotopy class depends only on the homotopy type of X. Let SτM denote the spherical fibration of the topological tangent bundle τM. If f : M X is a homotopy equivalence, then the spherical fibration f^{∗}σX ⊕ SτM is stably fiber homotopically trivial. To put it differently, f^{∗}σX is the stable inverse of the spherical fibration Sτ(M) of the tangent bundle in the fiber homotopy category. Let σ_{T} X be a stable inverse of σX. Then f^{∗}σ_{T} X is stably fiber homotopy equivalent to Sτ(M); in particular, the stable fiber homotopy type of Sτ(M) is a homotopy invariant of M. We used this fact already in Section 8. To make things more concrete, let us say that σ_{T} X is an N-spherical fibration, for N > 2 dim(M). Then f : M X induces a bundle map τM ⊕ \underline{ℝ}^{N−dim(M)} f^{∗}σ_{T} X which is a fiber homotopy equivalence. Such a map is called a TOP reduction of f^{∗}σ_{T} X. Two reductions are called equivalent if they differ by a fiber homotopy equivalence; the set of all stable TOP reductions of a spherical fibration ϕ is denoted RTOP(ϕ). One can show that every element of S_{TOP}(X) yields a well-defined reduction of σ_{T} (X); this correspondence is injective, and we obtain an injection S_{TOP}(X) RTOP(σ_{T} X). We call T_{TOP}(X) = RTOP(σ_{T} X) the set of tangential invariants of X. (Most texts consider normal invariants instead of tangential invariants. Since we are working in the stable category, the difference is merely the sign. Kirby-Siebenmann [33] use tangential invariants.)

Given a spherical fibration ϕ which admits a TOP reduction, it can be shown that the abelian group [X; G/TOP] acts regularly on RTOP(ϕ); thus there is a bijection of sets

T_{TOP}(X) ≅ [X; G/TOP].

Now we can state the surgery classification of manifolds of a given homotopy type. So let X be a 1-connected finite Poincaré duality complex of formal dimension n ≥ 5. There exists an abelian group P_{n} and a map θ : T_{TOP}(X) P_{n}, such that S_{TOP}(X) is precisely the preimage θ^{−1}(0); in other words, there is an exact sequence of sets

[figure: S_{TOP}(X); ⊂; inj; T_{TOP}(X); θ; P_{n}]

..................................

[X; G/TOP]

the surgery exact sequence. The dotted arrow is in general not a group homomorphism (and T_{TOP}(X) has no canonical group structure). If n = 4k ≥ 8 (the case we are interested in), then P_{n} ≅ ℤ and θ is connected to the L-genus and the signature as

θ(ξ) = \frac{1}{8}(⟨L_{4k}(ξ), [X]⟩ − Sig(X)).

In other words, a bundle ξ represents a homotopy manifold structure for X if and only if ξ satisfies Hirzebruch’s signature theorem.

Using these techniques, the proof of Theorem 9.2 is accomplished by the following steps.

Step 1. Every 1-connected Poincaré duality complex P (of formal dimension 2m ≥ 5) which has the same homology as a projective plane is homotopy equivalent to one of our models M(ξ).

Step 2. Let ϕ be a stable spherical fibration over a finite, 1connected CW-complex X with the property that H_{k}(X) = 0 for k ≢ 0 (mod 4). Then the Pontrjagin character ph injects RTOP(ϕ) into H^{•}(X; ℚ). In particular, ph : T_{TOP}(P) H^{•}(P; ℚ) is an injection.

Step 3. For m = 4, 8, we show that the elements of S_{TOP}(P) are completely determined by their Pontrjagin classes p_{m}, and that all possibilities for the p_{m} are realized through our models M(ξ).

Step 4. We determine Aut(P) and M_{TOP}(P).

9.8. In the remainder of this section, we carry out Steps 1–4. Let P be a 1-connected Poincaré duality complex of formal dimension 2m which is like a projective plane, for m = 4, 8. So H_{k}(P) ≅ ℤ for m = 0, 1, 2. We fix a map

s : 𝕊^{m} P

representing a generator of π_{m}(P) ≅ H_{m}(P) ≅ ℤ. Note also that P has a preferred orientation [P] — the class dual to y_{m}^{2}, where y_{m} ∈ H^{m}(P) ≅ ℤ is any generator. Thus, any homotopy equivalence automatically preserves fundamental classes. By Wall [63] Proposition 4.1, we may assume that

P = 𝕊^{m} ∪_{α} e^{2m}

is a 2-cell complex, and that m = 4, 8, see 10.1.

Proof for Step 1. This is just Theorem 8.4. However, we can say a bit more: the results in Section 8 show that the homotopy type of P is uniquely determined by the stable weak type of the spherical fibration s^{∗}σ_{T} P, i.e., by stable type of the pair of spherical fibrations {s^{∗}σ_{T} P, i^{∗}s^{∗}σ_{T} P }, where i : 𝕊^{m} 𝕊^{m} is any map of degree −1.

Proof for Step 2. Let ϕ be a stable spherical fibration over a connected CW-complex X, with stable classifying map c : X BG. A (stable) TOP-reduction of c is a lift C

BTOP

[figure: C]
[figure: X; c; BG,; f_{G}^{TOP}]

with f_{G}^{TOP} ∘ C = c. Two such lifts C_{0}, C_{1} are equivalent if there exists a homotopy C : X ×[0, 1] BTOP with c = f_{G}^{TOP} ∘C_{t} for all t ∈ [0, 1], i.e., if the homotopy is constant when projected to BG. The set of equivalence classes of lifts of c is denoted RTOP(ϕ). If c is the constant map (and thus ϕ = \underline{0} is trivial), we obtain a bijection RTOP(\underline{0}) ≅ [X; G/TOP]. For η ∈ RTOP(ϕ) and ζ ∈ RTOP(ψ) we have η ⊕ ζ ∈ RTOP(ϕ ⊕ ψ). This establishes a bijection

RTOP(\underline{0})  RTOP(ϕ)
η ⟼ η ⊕ ζ

see Wall [64] Sec. 10, p. 113. Thus, we can identify RTOP(ϕ) with [X; G/TOP], provided that RTOP(ϕ) ≠ ∅. Note however that RTOP(ϕ) has in general no natural group structure; rather, the abelian group [X; G/TOP] acts regularly on this set.

In general, different elements of RTOP(ϕ) can be equivalent when viewed as stable bundles. From the homotopy viewpoint, this is due to the fact that two lifts C, C′ can be homotopic without being fiber homotopic. We prove now that under certain conditions on X, the map RTOP(ϕ) \overline{}[X, BTOP] is injective.

Proposition 9.9. Let X be a finite 1-connected CW-complex. Assume that H_{k}(X) = 0 for all k ≢ 0 (mod 4). Then the natural map [X; G/TOP] \overline{}[X; BTOP] is injective.

Before we start with the proof, we note the following. By Theorem 4.2, we have isomorphisms K̃O(𝕊^{k}) ⊗ ℚ { \atop KTOP}(𝕊^{k}) ⊗ ℚ for all k ≥ 0. By a well-known comparison theorem for half-exact cofunctors, see Dold [12] Ch. 7, or by the Atiyah-Hirzebruch spectral sequence, see Hilton [22] Ch. 3, this implies that

K̃O(Y) ⊗ ℚ ^{≅} { \atop KTOP}(Y) ⊗ ℚ

is a natural isomorphism of homotopy functors for every finite connected CW-complex Y. We combine this with the Pontrjagin character ph, which is (by the same comparison theorem for half-exact cofunctors and by Bott Periodicity, see Section 4) rationally an isomorphism

K̃O(Y) ⊗ ℚ ph ; ≅ H̃^{4•}(Y; ℚ)

to obtain an isomorphism

{ \atop KTOP}(Y) ⊗ ℚ ph ; ≅ H̃^{4•}(Y; ℚ)

which we also denote by ph.

Proof of Proposition 9.9. It clearly suffices to show that the natural map

[X; G/TOP] { \atop KTOP}(X) ⊗ ℚ

is an injection. First we note that this is true in general for X = 𝕊^{4k+t}, for t = −1, 0, 1 (note that π_{4k±1}(G/TOP) = 0, by Theorem 4.1). Both [−; G/TOP] and { \atop KTOP}(−) ⊗ ℚ are half-exact cofunctors, so injectivity holds also for a wedge of spheres X = ⋁_{1}^{r} 𝕊^{4k+t} (this is just the additivity of half-exact cofunctors).

For a general complex X as in the claim of Proposition 9.9, we proceed by induction. By standard obstruction theory, we may assume that X is complex whose cells all have dimensions divisible by 4, i.e., that X^{(4k)} = X^{(4k+1)} = X^{(4k+2)} = X^{(4k+3)} for all k, see Wall [63] Proposition 4.1. So suppose that X = X^{(4k)}, and that A = X^{(4k−1)} = X^{(4k−4)}. The long exact sequence of the pair (X, A) shows that A also satisfies the hypothesis of Proposition 9.9, and by induction, we may assume that the conclusion of the proposition holds for A. Now consider the Puppe sequence

⋁_{r} 𝕊^{4k−1}  A  X  ⋁_{r} 𝕊^{4k}  SA  ⋯

Note that { \atop KTOP}(SA) ⊗ ℚ ≅ H̃^{4•}(SA; ℚ) = 0, and that similarly { \atop KTOP}(⋁_{r} 𝕊^{4k−1}) ⊗ ℚ = 0. We thus obtain a diagram

[figure: 0; [A; G/TOP]; [X; G/TOP]; [⋁_{r} 𝕊^{4k}; G/TOP]; [SA; G/TOP]]

[figure: 0; { \atop KTOP}(A) ⊗ ℚ; inj; { \atop KTOP}(X) ⊗ ℚ; { \atop KTOP}(⋁_{r} 𝕊^{4k}) ⊗ ℚ; inj; 0; inj]

and this implies by the Five-Lemma that [X; G/TOP] { \atop KTOP}(X)⊗ ℚ is injective, see, e.g., Eilenberg-Steenrod [16] Lemma 4.4. This finishes the proof of Proposition 9.9. q.e.d.

Corollary 9.10. Let ϕ be a spherical fibration over a finite 1connected CW-complex X, with H_{k}(X) = 0 for all k ≢ 0 (mod 4), and assume that RTOP(ϕ) ≠ ∅. Then RTOP(ϕ) injects into { \atop KTOP}(X) ⊗ ℚ and, via the Pontrjagin character, into H̃^{4•}(X; ℚ).

Proof. Let ζ ∈ RTOP(ϕ) be a stable bundle. By Proposition 9.9, the Pontrjagin character injects RTOP(\underline{0}) into H̃^{4•}(X; ℚ). The diagram

RTOP(\underline{0}) \overline{\begin{matrix}[η ⟼ η + ζ] \\ bij \end{matrix}} RTOP(ϕ)

[figure: H̃^{4•}(X; ℚ); inj ph; \begin{matrix}[x ⟼ x + ph(ζ)] \\ bij \end{matrix}; H̃^{4•}(X; ℚ); ph]

commutes, and the claim follows. q.e.d.

Corollary 9.11. Let X be a finite 1-connected Poincaré duality complex, and assume that H_{k}(X) = 0 for all k ≢ 0 (mod 4). Then the Pontrjagin character ph injects the set T_{TOP}(X)of tangential invariants into H̃^{4•}(X; ℚ). q.e.d.

This corollary applies in particular to our Poincaré duality complex P of formal dimension 2m, for m = 4, 8. Note also the following. If η, ζ ∈ RTOP(ϕ) are elements with the same total Pontrjagin class, p(η) = p(ζ), then clearly ph(η) = ph(ζ). Therefore, the total Pontrjagin class p induces also an injection of RTOP(ϕ) into H̃^{4•}(X; ℚ).

Proof for Step 3. We use the same symbols P, y_{m}, y_{2m}, s as in 9.8. Since Sig(P) = 1, a stable TOP-bundle reduction ζ ∈ T_{TOP}(P) represents a homotopy manifold structure for P if and only if

L_{2m}(ζ) = y_{2m}^{2}.

Also, we see from the formula for the L-genus 4.6 that there are rational numbers c_{m}, d_{m} (depending only on m) such that

p_{2m}(η) = c_{m}L_{2m}(η) + d_{m}p_{m}(η)^{2}

for any stable bundle η over P.

Lemma 9.12. The map T_{TOP}(P) H^{m}(P; ℚ), ζ ⟼ p_{m}(ζ) is an injection when restricted to S_{TOP}(P) ⊆ T_{TOP}(P).

Proof. Let η, ζ be elements in T_{TOP}(X) representing homotopy manifold structures, so L_{2m}(η) = y_{m}^{2} = L_{2m}(ζ). If p_{m}(η) = p_{m}(ζ), then p_{2m}(η) = p_{2m}(ζ) by the formula above. Thus p(η) = p(ζ), whence ph(η) = ph(ζ), and therefore η = ζ by Corollary 9.10. q.e.d.

Now we prove that our set of model manifolds realizes all elements in S_{TOP}(P). Let ϕ = s^{∗}σ_{T} P.

Lemma 9.13. Let ζ ∈ RTOP(ϕ). Then there exists an ℝ^{m}-bundle ξ over 𝕊^{m} with absolute Euler number |e| = 1 which is stably equivalent to ζ.

Proof. We have w_{m}(ζ) = x mod 2 (because w_{m}(M_{r,s}) = y_{m} mod 2 for any model). From the split exact sequence in Lemma 6.2 we see that we can find an ℝ^{m}-bundle ξ over 𝕊^{m} with w_{m}(ξ) = x mod 2, and hence with any odd absolute Euler number. q.e.d.

Corollary 9.14. For m = 4, 8, all elements of S_{TOP}(X) are realized as model manifolds M(ξ).

Proof. Given a stable bundle η ∈ T_{TOP}(P) representing a homotopy manifold structure M X in S_{TOP}(X), we can find by Lemma 9.13 an oriented ℝ^{m}-bundle ξ over 𝕊^{m} which is stably equivalent to s^{∗}η, with absolute Euler number |e| = 1. Then the model manifold M(ξ) is homotopy equivalent to P by the remark in Step 1, because the homotopy types of P and M(ξ) are determined by s^{∗}σ_{T} P and s_{0}^{∗}σ_{T} M(ξ), respectively. Composing the homotopy equivalence M(ξ) ≃ P with a self-homeomorphism of M(ξ) induced by a homeomorphism of degree −1 of 𝕊^{m}, if necessary, we obtain a homotopy commutative diagram

𝕊^{m}  \overline{s} P
[figure: s_{0}; f; ≃]
M(ξ).

By Lemma 9.12, M(ξ) is precisely the homotopy manifold structure on P represented by η, because f^{•}p_{m}(M) = p_{m}(η), so M ≅ M(ξ). q.e.d.

This finishes Step 3.

Proof for Step 4. We proved already in Proposition 3.4 that M(ξ) ≅ M(ξ′) if and only if ξ and ξ′ are weakly equivalent. The group Aut(M(ξ)) is cyclic of order two by Lemma 10.5, and this finishes the classification for m = 4, 8.

10. Appendix: Homotopy classification

In this section, all maps and homotopies are assumed to preserve base points.

10.1. Suppose that X is a 1-connected Poincaré duality complex of formal dimension n, with H_{•}(X) ≅ ℤ^{3} (9.3). Let μ ∈ H_{n}(X) denote the fundamental class. We have H_{0}(X) ≅ ℤ ≅ H_{n}(X), so H_{m}(X) ≅ ℤ for some number 1 < m < n. By the universal coefficient theorem, see Spanier [55] Ch. 5.5 Theorem 4, we have H^{j}(X) ≅ H_{j}(X) for all j. From Poincaré duality, we see that n = 2m, and that the map

H^{m}(X) ⊗ H^{m}(X)  ℤ,  u ⊗ v ⟼ ⟨u⌣v, μ⟩

is a duality pairing. Thus m is even, and H^{•}(X) ≅ ℤ[y_{m}]/(y_{m}^{3}), for some generator y_{m} ∈ H^{m}(X). By Wall [63] Proposition 4.1, the CW-complex X is homotopy equivalent to a 2-cell complex,

X ≃ X_{α} = 𝕊^{m} ∪_{α} e^{2m},

for some attaching map α : 𝕊^{2m−1} 𝕊^{m}. By Adams-Atiyah [2] Theorem A, this implies that m = 2, 4, 8. Note also that X_{α} has a preferred orientation μ, the dual of y_{m}^{2}.

10.2. We wish to determine the number of homotopy types of such complexes X. The structure of the cohomology ring of X_{α} implies that α, viewed as an element of π_{2m−1}(𝕊^{m}), has Hopf invariant h(α) = ±1, see Adams-Atiyah [2] and Husemoller [30] Ch. 20.10. Note also that the homotopy type of X_{α} is not changed if α is replaced by a map homotopic to α, see Milnor [45] Lemma 3.6. Let W_{m} denote the set of all homotopy types of 1-connected CW-complexes as above. Let H_{m}^{±1} ⊆ π_{2m−1}(𝕊^{m}) denote the set of all elements of Hopf invariant ±1. We thus have a well-defined surjection

π_{2m−1}(𝕊^{m}) ⊇ H_{m}^{±1} W_{m},

sending α ∈ H_{m}^{±1} to the homotopy type of X_{α}. By Adams-Atiyah [2] Theorem A, the set W_{m} is empty unless m = 2, 4, 8.

Next we note the following. If c : 𝕊^{2m−1} 𝕊^{2m−1} is an involution of degree −1, then there is a homeomorphism X_{α} ≅ X_{α∘c}. Since α ∘ c represents −α ∈ π_{2m−1}(𝕊^{m}), we have a homotopy equivalence

X_{α} ≃ X_{−α}.

Each element in W_{m} is thus represented by a element α ∈ π_{2m−1}(𝕊^{m}) with Hopf invariant h(α) = 1, i.e., H_{m}^{+} surjects onto W_{m}.

For m = 2, we are done: π_{3}(𝕊^{2}) ≅ ℤ, see Toda [60] p. 186, so H_{2}^{±} has exactly two elements, and W_{2} consists of precisely one homotopy type, the complex projective plane ℂP^{2}.

Lemma 10.3. There is exactly one homotopy type in W_{2}.

10.4. It remains to consider the cases m = 4, 8. Similarly as above, if g : 𝕊^{m} 𝕊^{m} is an involution of degree −1, then there is a homeomorphism X_{α} ≅ X_{g∘α}, so

X_{α} ≃ X_{g_{#}(α)}.

The homotopy equivalences

X_{α} ≃ X_{−α} ≃ X_{g_{#}(α)} ≃ X_{−g_{#}(α)}

are in fact the only homotopy equivalences which occur between these 2-cell complexes. For suppose that f : X_{α} ^{≃} X_{β} is a homotopy equivalence. We may assume that f is a cellular map, see Whitehead [66] Ch. II Theorem 4.5,

f : (X_{α}, 𝕊^{m}) (X_{β}, 𝕊^{m}),

and then f restricts to a map of degree ±1 on the m-skeleton 𝕊^{m}. Also, we have the Hurewicz isomorphism

H_{2m}(𝕊^{2m}) = H_{2m}(X_{α}/𝕊^{m}) ≅ H_{2m}(X_{α}, 𝕊^{m}) ≅ π_{2m}(X_{α}, 𝕊^{m}) ≅ ℤ;

a canonical generator α̂ of this group π_{2m}(X_{α}, 𝕊^{m}) is given by the attaching map,

(e^{2m}, 𝕊^{2m−1}) ^{α̂} (X_{α}, 𝕊^{m}) with ∂α̂ = α.

From the homotopy exact sequence

[figure: π_{2m}(𝕊^{m}); π_{2m}(X_{α}); π_{2m}(X_{α}, 𝕊^{m}); ∂; π_{2m−1}(𝕊^{m}); π_{2m}(𝕊^{m}); π_{2m}(X_{β}); π_{2m}(X_{β}, 𝕊^{m}); ∂; π_{2m−1}(𝕊^{m})]

and the Five-Lemma, we see that f_{#}(α̂) = ±β̂, and so f_{#}(α) = ±β. We have seen above that we may replace X_{β} by X_{−β}; thus, we may assume that f_{#}(α) = β. If f restricts to a map of degree 1 on 𝕊^{m}, then f_{#}(α) = α = β.

Lemma 10.5. The group of self-equivalences Aut(X_{α}) is cyclic of order 2; it coincides with the group of graded ring automorphisms of the cohomology ring ℤ[y_{m}]/(y_{m}^{3}).

10.6. So the remaining problem is to determine the relation between α and g_{#}(α), where g : 𝕊^{m} 𝕊^{m} is a map of degree −1. Towards this end, we consider the EHP -sequence of 𝕊^{n} for the values n = m − 1, m,

π_{k}(𝕊^{n}) ^{E} π_{k+1}(𝕊^{n+1}) ^{H} π_{k+1}(𝕊^{2n+1}) ^{P} π_{k−1}(𝕊^{n}) ^{E}

This sequence is exact for k ≤ 3n − 2, see Whitehead [66] Ch. XII Theorem 2.2. Here, E is the suspension and H is the generalized Hopf invariant. Let ι_{j} = id_{𝕊^{j}} denote the canonical generator of π_{j}(𝕊^{j}). For ρ ∈ π_{2n+1}(𝕊^{n}), one has H(ρ) = h(ρ) · ι_{2n+1}; see Whitehead [66] for a comparison between the various definitions of Hopf invariants.

10.7. Consider the diagram

0

[figure: π_{2m−1}(𝕊^{2m−1}); E^{2} ; ≅; π_{2m+1}(𝕊^{2m+1})]
[figure: ·2]
[figure: 0; π_{2m−2}(𝕊^{m−1}); E; π_{2m−1}(𝕊^{m}); P; H; π_{2m−1}(𝕊^{2m−1}); 0]

[figure]

π_{2m}(𝕊^{m+1}) \begin{matrix}E \\  \end{matrix}

0.

The middle row is split and short exact: π_{2m−1}(𝕊^{2m−1}) is infinite cyclic (whence the splitting) and the Hopf invariant H is by assumption onto. The map P : π_{2m}(𝕊^{2m−1}) π_{2m−2}(𝕊^{m−1}) can be characterized by P E^{2}(ρ) = [ι_{m−1}, ι_{m−1}] ∘ ρ, see Whitehead [66] Ch. XII Theorem 2.4. But 𝕊^{m−1} is an H-space for m = 4, 8, so [ι_{m−1}, ι_{m−1}] = 0, see Whitehead [66] Ch. X Corollary 7.8. From the EHP sequence, we see that π_{2m−2}(𝕊^{m−1}) ^{E} π_{2m−1}(𝕊^{m}) is an injection.

The middle column is also short exact: from Freudenthal’s Suspension Theorem we have that E : π_{2m−1}(𝕊^{m}) π_{2m}(𝕊^{m+1}) is an epimorphism, see Whitehead [66] Ch. VII Theorem 7.13. To see that P : π_{2m+1}(𝕊^{2m+1}) π_{2m−1}(𝕊^{m}) is injective, note that P E^{2}(ι_{2m−1}) = [ι_{m}, ι_{m}] by Whitehead [66] Ch. XII Theorem 2.4. But

H([ι_{m}, ι_{m}]) = 2ι_{2m−1},

see Whitehead [66] Ch. XI Theorem 2.5. Thus, P is injective on this infinite cyclic group.

10.8. So suppose that g : 𝕊^{m} 𝕊^{m} has degree −1. Then

g_{#}([ι_{m}, ι_{m}]) = [g_{#}(ι_{m}), g_{#}(ι_{m})] = [−ι_{m}, −ι_{m}] = [ι_{m}, ι_{m}], whence

H(g_{#}(ξ)) = H(ξ) for all ξ ∈ π_{2m−1}(𝕊^{m}).

Now let ρ ∈ π_{2m−2}(𝕊^{m−1}). This group is finite by Serre’s finiteness result for odd spheres, see Spanier [55] Ch. 9.7 Theorem 7; via E, we can identify it with the torsion group of π_{2m−1}(𝕊^{m}). The double suspension E^{2} injects this group into the stable group π_{2m}(𝕊^{m+1}) = π_{m−1}^{s}(𝕊^{0}). In the graded ring π_{•}^{s}(𝕊^{0}), composition is commutative, see Whitehead [66] Ch. XII; thus we have E(g_{#}(Eρ)) = −E^{2}(ρ), whence g_{#}(E(ρ)) = −E(ρ). The involution g_{#} thus changes the signs of the elements of the torsion group of π_{2m−1}(𝕊^{m}).

The stable groups π_{m−1}^{s}(𝕊^{0}) are cyclic of order 24 and 240, for m = 4, 8, see Toda [60] p. 186. The image of the double suspension E^{2} is also cyclic and has index 2 in this group. Thus, if ξ ∈ π_{2m−1}(𝕊^{m}) is an element with Hopf invariant 1, then the element E(ξ) together with the cyclic group E^{2}(π_{2m−2}(𝕊^{m−1})) generates π_{2m+2}(𝕊^{m+1}) ≅ π_{m−1}^{s}(𝕊^{0}). Therefore, we can find an element η_{m} ∈ π_{2m−1}(𝕊^{m}) with Hopf invariant h(η_{m}) = 1 whose suspension E(η_{m}) generates π_{2m}(𝕊^{m+1}). Let δ_{m} = 2η_{m} − [ι_{m}, ι_{m}]; then E(δ_{m}) = 2 · E(η_{m}) generates the cyclic group E^{2}(π_{2m−2}(𝕊^{m−1})) (the suspension of the Whitehead product [ι_{m}, ι_{m}] vanishes, see Whitehead [66] Ch. X Theorem 8.20), so δ_{m} generates the torsion group of π_{2m−1}(𝕊^{m}). Put g_{#}(η_{m}) = η_{m} + r · δ_{m}. Now

−δ_{m} = g_{#}(δ_{m}) = g_{#}(2η_{m} − [ι_{m}, ι_{m}])
= 2η_{m} + 2r · δ_{m} − [ι_{m}, ι_{m}] = (1 + 2r) · δ_{m},

whence 2(1 + r)δ_{m} = 0. This leaves two possibilities for r, for m = 4, 8. But due to the commutativity of π_{•}^{s}(𝕊^{0}), we have also E(g_{#}(η_{m})) = −E(η_{m}) = E(η_{m}) + r · E(δ_{m}) = (1 + 2r)E(η_{m}), which implies that r = −1, i.e., that

g_{#}(η_{m}) = η_{m} − δ_{m}.

We have proved the following result.

Proposition 10.9. The number of homotopy types in W_{m} is 1, 6, 60, for m = 2, 4, 8.

This is a complete homotopy classification of manifolds and complexes which are like projective planes, and also the end of this paper.

References

[1] J.F. Adams, On the groups J(X)–IV, Topology 5 (1966) 21–71, MR 33 #6628, Zbl 0145.19902.

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

[3] M.F. Atiyah & F. Hirzebruch, Vector bundles and homogeneous spaces, Proc. Symp. Pure Math. 3 (1961) 7–38, MR 25 #2617, Zbl 0108.17705.

[4] R. Bott, The stable homotopy of the classical groups, Ann. Math. 70 (1959) 313–227,

[5] R. Bott, The periodicity theorem for the classical groups and some of its applications, Adv. Math. 4 (1970) 353–411, MR 41 #4533, Zbl 0231.55010.

[6] U. Brehm & W. Kühnel, 15-vertex triangulations of an 8-manifold, Math. Ann. 294 (1992) 167–193, MR 94e:57033, Zbl 0734.57017.

[7] S. Breitsprecher, Projektive Ebenen, die Mannigfaltigkeiten sind, Math. Z. 121 (1971) 157–174, MR 43 #6935, Zbl 0229.50021.

[8] G. Brumfiel, On the homotopy groups of BPL and PL/O, Ann. Math. 88 (1968) 291–311, MR 38 #2775, Zbl 0162.27302.

[9] T. Buchanan, Zur Topologie der projektiven Ebenen über reellen Divisionsalgebren, Geom. Ded. 8 (1979) 383–393, MR 81m:51023, Zbl 0414.51008.

[10] H. Cartan & S. Eilenberg, Homological algebra, Princeton University Press, Princeton, New Jersey, 1956, MR 17,1040e, Zbl 0075.24305.

[11] A. Dold, Partitions of unity in the theory of fibrations, Ann. Math. 78 (1963) 223–255, MR 27 #5246, Zbl 0203.25402.

[12] A. Dold, Halbexakte Homotopiefunktoren, Springer LNM, 12, Springer-Verlag, 1966, MR 33 #6622, Zbl 0136.00801.

[13] A. Dold, Lectures on algebraic topology, Springer Verlag, Berlin Heidelberg New York, 1972, MR 54 #3685, Zbl 0234.55001.

[14] J. Eells & N.H. Kuiper, Manifolds which are like projective planes, Publ. Math. I.H.E.S. 14 (1962) 5–46, MR 26 #3075, Zbl 0109.15701.

[15] J. Eells & N.H. Kuiper, An invariant for certain smooth manifolds, Ann. Mat. Pura Appl. 60 (1962) 93–110, MR 27 #6280, Zbl 0119.18704.

[16] S. Eilenberg & N. Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, New Jersey, 1952, MR 14,398b, Zbl 0047.41402.

[17] R. Engelking, Theory of dimensions, finite and infinite, Heldermann, Lemgo, 1995, MR 97j:54033 zbl0872.54002.

[18] A.T. Fomenko, D.B. Fuchs & V.L. Gutenmacher, Homotopic topology, Akadémiai Kiadó, Budapest, 1986, MR 88f:55001, Zbl 0615.55001.

[19] M.H. Freedman, The topology of four-dimensional manifolds, J. Differential Geometry 17 (1982) 357–453, MR 84b:57006, Zbl 0528.57011.

[20] M.H. Freedman & F. Quinn, Topology of 4-manifolds, Princeton Univ. Press, Princeton, 1990, MR 94b:57021, Zbl 0705.57001.

[21] O. Hanner, Some theorems on absolute neighborhood retracts, Arkiv Mat. 1 (1951) 389–408, MR 13,266d, Zbl 0042.41102.

[22] P. Hilton, General cohomology theory and K-theory, Cambridge Univ. Press, Cambridge, 1971, MR 44 #5953.

[23] M.W. Hirsch, Obstruction theories for smoothing manifolds and maps, Bull. Amer. Math. Soc. 69 (1963) 352–356, MR 26 #6980, Zbl 0112.14602.

[24] F. Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, zweite erg. Auflage, Springer-Verlag Berlin Göttingen Heidelberg, 1962, MR 25 #1155, Zbl 0101.38301.

[25] F. Hirzebruch, A Riemann-Roch theorem for differentiable manifolds, Sém. Bourbaki Exp., 177, Paris, 1959; reprinted in: F. Hirzebruch, Ges. Abh. Band I, 481–495, Springer Verlag, Berlin Heidelberg New York, 1987, MR 1 603 462.

[26] P. Holm, The microbundle representation theorem, Acta math. 117 (1967) 191– 213, MR 34 #8427, Zbl 0148.17502.

[27] S.-T. Hu, Homotopy theory, Academic Press, 1959, MR 21 #5186, Zbl 0088.38803.

[28] S.-T. Hu, Theory of retracts, Wayne State University Press, 1965, MR 31 #6202, Zbl 0145.43003.

[29] W. Hurewicz & H. Wallman, Dimension theory, Princeton University Press, Princeton, 1948, MR 3,312b, Zbl 0036.12501.

[30] D. Husemoller, Fibre bundles, 3rd ed., Springer Verlag, 1994, MR 94k:55001, Zbl 0794.55001.

[31] P. Kahn, A note on topological Pontrjagin classes and the Hirzebruch index formula, Illinois J. Math. 16 (1972) 234–256, MR 45 #1183, Zbl 0238.57005.

[32] M. Kervaire & J. Milnor, Groups of homotopy spheres, I, Ann. Math. 77 (1963) 504–537, MR 26 #5584, Zbl 0115.40505.

[33] R.C. Kirby & L.C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press, Princeton, New Jersey, 1977, MR 58 #31082, Zbl 0361.57004.

[34] J.R. Klein, Poincaré duality spaces, 135–166, in ‘Surveys on surgery theory’, Vol. I, S. Cappell, A. Ranicki & J. Rosenberg eds., Princeton Univ. Press, 2000, MR 2001b:57053, Zbl 0946.57027.

[35] N. Knarr, The nonexistence of certain topological polygons, Forum Math. 2 (1990) 603–612, MR 92a:51018, Zbl 0711.51002.

[36] L. Kramer, The topology of smooth projective planes, Arch. Math. 63 (1994) 85–91, MR 95k:51021, Zbl 0831.51011.

[37] L. Kramer, The point and line space of a compact projective plane are homeomorphic, Math. Z. 234 (2000) 83–102, MR 2001c:55009, Zbl 0966.51008.

[38] M. Kreck, A guide to the classification of manifolds, 121–134, in ‘Surveys on surgery theory’, Vol. I, S. Cappell, A. Ranicki & J. Rosenberg eds., Princeton Univ. Press, 2000, MR 2001b:57061, Zbl 0947.57001.

[39] H.B. Lawson & M.-L. Michelsohn, Spin geometry, Princeton University Press, Princeton, New Jersey, 1989, MR 91q:53001, Zbl 0688.57001.

[40] R. Löwen, Topology and dimension of stable planes: On a conjecture of H. Freudenthal, J. Reine Angew. Math. 343 (1983) 108–122, MR 85b:57020, Zbl 0524.57011.

[41] I. Madsen & R.J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Princeton University Press, Princeton, New Jersey, 1979, MR 81b:57014, Zbl 0446.57002.

[42] J. McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge University Press, Cambridge, 2001, MR 2002c:55027, Zbl 0959.55001.

[43] J. Milnor, Microbundles–I, Topology 3(1) (1964) 53–80, MR 28 #4553b, Zbl 0124.38404.

[44] J.W. Milnor, On characteristic classes for spherical fibre spaces, Comm. Math. Helv. 43 (1968) 51–77, MR 37 #2227, Zbl 0153.53403.

[45] J.W. Milnor & J.D. Stasheff, Morse Theory, Princeton University Press, Princeton, New Jersey, 1963, MR 29 #634, Zbl 0108.10401.

[46] J.W. Milnor & J.D. Stasheff, Characteristic classes, Princeton University Press, Princeton, New Jersey, 1974, MR 55 #13428, Zbl 0298.57008.

[47] H.M.A. Newman, The engulfing theorem for topological manifolds, Ann. Math. 84 (1966) 555–571, MR 34 #3557, Zbl 0166.19801.

[48] T. Okabe, The existence of topological Morse function and its application to topological h-cobordism theorem, J. Fac. Sci. Univ. Tokyo Sect. IA Math 18 (1971) 23–35, MR 45 #2716, Zbl 0214.22401.

[49] F. Quinn, Ends of maps, I, Ann. Math. 110 (1979) 275–331, MR 82k:57009, Zbl 0394.57002.

[50] Y.B. Rudyak, On Thom spectra, orientability, and cobordism, Springer Verlag, 1998, MR 99f:55001, Zbl 0906.55001.

[51] J. Rosenberg, C^{∗}-algebras, positive scalar curvature, and the Novikov conjecture – III, Topology 25 (1986) 319–336, MR 88f:58141, Zbl 0605.53020.

[52] H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen, & M. Stroppel, Compact projective planes, de Gruyter Expositions in Mathematics, 21, Berlin, 1995, MR 79b:51009, Zbl 0851.51003.

[53] N. Shimada, Differentiable structures on the 15-sphere and Pontrjagin classes of certain manifolds, Nagoya Math. J. 12 (1957) 59–69, MR 20 #2715, Zbl 0145.20303.

[54] S. Smale, Generalized Poincaré’s conjecture in dimensions greater than four, Ann. Math. 74 (1961) 391–406, MR 25 #580, Zbl 0099.39202.

[55] E.H. Spanier, Algebraic topology, Springer, 1966, MR 35 #1007, Zbl 0145.43303.

[56] M. Spivak, Spaces satisfying Poincaré duality, Topology 6 (1967) 77–101, MR 35 #4923, Zbl 0185.50904.

[57] N.E. Steenrod, Topology of fibre bundles, Princeton Univ. Press, Princeton, 1951, MR 12,522b, Zbl 0054.07103.

[58] S. Stolz, Hochzusammenhängende Mannigfaltigkeiten und ihre Ränder, Springer LNM, 1116, Springer Verlag, 1985, MR 88f:57061, Zbl 0561.57021.

[59] S. Stolz, Simply connected manifolds of positive scalar curvature, Ann. Math. 136 (1992) 511–540, MR 99i:57033, Zbl 0784.53029.

[60] H. Toda, Composition methods in homotopy groups of spheres, Princeton Univ. Press, Princeton, 1962, MR 26 #777, Zbl 0101.40703.

[61] K. Varadarajan, Span and stably trivial bundles, Pacific J. Math. 60 (1975) 277– 287, MR 54 #13931, Zbl 0319.55026.

[62] C.T.C. Wall, Classification of (n − 1)-connected 2n-manifolds, Ann. Math. 75 (1962) 163–189, MR 26 #3071, Zbl 0218.57022.

[63] C.T.C. Wall, Finiteness conditions for CW-complexes, Ann. Math. 81 (1965) 56–69, MR 30 #1515, Zbl 0152.21902.

[64] C.T.C. Wall, Surgery on compact manifolds, 2nd ed., Mathematical surveys and monographs, 69, Amer. Math. Soc., 1999, MR 2000a:57089, Zbl 0935.57003.

[65] C. Weber, Quelques théoremes bien connus sur les A.N.R et les C.W. complexes, Enseign. Math. 13 (1968) 211–222, MR 40 #6533, Zbl 0157.30002.

[66] G.W. Whitehead, Elements of homotopy theory, Springer-Verlag New York Berlin Heidelberg, 1978, MR 80b:55001, Zbl 0406.55001.

[67] R.E. Williamson, Jr., Cobordism of combinatorial manifolds, Ann. Math. 83 (1966) 1–33, MR 32 #1715, Zbl 0137.42901.

Mathematisches Institut Universität Würzburg

Am Hubland D-97074 Würzburg

Germany