j. differential geometry 59 (2001) 1-14

INDEX THEORY WITH BOUNDED GEOMETRY, THE UNIFORMLY FINITE Â CLASS, AND INFINITE

CONNECTED SUMS

KEVIN WHYTE

Abstract

We prove a vanishing theorem in uniformly finite homology for the Â genus of a complete spin manifold of bounded geometry and non-negative scalar curvature. This theorem is then applied to obstruct the existence of such metrics for some infinite connected sums, giving a converse to a theorem of Block and Weinberger.

Introduction

A manfold of bounded geometry is a Riemannian manifold with bounds on its curvature tensor and its derivatives, and on the injectivity radius. The natural equivalence relation is diffeomorphism with bounded distortion. Unless otherwise stated, all manifolds in the paper are assumed of this type.

These definitions are designed to reflect the restrictions imposed on a noncompact manifold which is controlled in some way by a compact manifold. The most common example of this is a covering of a compact manifold. Any metric on the base gives a metric of bounded geometry on the cover, and any two such metrics are bounded distortion equivalent. Similarly, leaves of foliations of compact manifolds have a canonical bounded distortion class of metric of bounded geometry.

We try to understand index theory for these manifolds, and in particular, questions of positive scalar curvature. Generally, the appropriate notion here is uniformly positive scalar curvature, meaning the scalar curvature is bounded away from zero from below. When we say that M admits a metric of positive scalar curvature, we mean that within the chosen bounded distortion class of metrics there is a metric of uniformly positive scalar curvature.

Received April 2, 2000

To understand positive scalar curvature we need an appropriate generalization of the Â class. One interesting feature which emerges is that this class lives in a non-Hausdorff homology group, and thus standard C^{∗} algebra methods do not apply. It turns out that to understand this class requires rather delicate spectral estimates for Dirac operators on the boundaries of certain compact submanifolds. The resulting theorems have unexpected applications to compact manifolds.

The motivation for this work comes from some interesting infinite connected sum examples studied in [5] and [16]. As these examples provide good motivation for the definition of the Â class we use, and are interesting on their own, we discuss them in some detail in the first section. The gap between the obstruction of [16] and the construction in [6] is a direct consequence of the non-Hausdorff problem mentioned above. Our theorems solve this problem, and provide a complete picture of these examples.

We would like to thank Shmuel Weinberger for bringing this problem to our attention, and for many helpful discussions. We would also like to thank the referee, without whom this paper would be substantially less readable.

1. Infinite Connected Sums

Let M be a manifold of bounded geometry and S a discrete subset of M. Given N, a compact manifold, we form a new manifold, M#_{S}N, by connected summing a copy of N at each point of S. For this manifold to have a well defined bounded distortion class of metrics of bounded geometry one needs that points of S are uniformly separated, meaning that for some ε > 0 any two distinct points of S are at least ε apart. Such a set is called uniformly discrete. One important example of this construction is for M the universal cover of some X and S an orbit of π_{1}(X). Then M#_{S}N is just a cover of X#N.

Now suppose M has positive scalar curvature. We wish to understand when M#_{S}N admits a metric of positive scalar curvature. We restrict our attention to the case of M and N spin, and N simply connected. If we are in the covering space situation, there is a positive scalar curvature metric invariant under the covering group if and only if Â(N) = 0. We are looking to generalize the Â class as an obstruction to positive scalar curvature, so we will typically assume Â(N) ≠ 0.

A natural first question is whether it is ever possible, keeping bounded geometry, to have a metric of positive scalar curvature in the presence of any such “obstructing” N. Examples showing this is possible are constructed in [6]. We sketch their construction below as it gives insight about the kind of obstructions that arise in our main theorem, Theorem 2.3.

Lemma 1.1 ([6]). Let M be noncompact, spin, and of uniformly positive scalar curvature, and N be compact, spin, and simply connected. The manifold M#N admits a metric of uniformly positive scalar curvature.

Proof. Choose a geodesic ray p in M. Let M̂ be the infinite connected sum of M with copies of N at p(2i) and copies of N (N with opposite orientation) at p(2i + 1), for every i. We view M̂ in two ways. First, we think of the copies of N as coming in pairs, at p(2i) and p(2i + 1). Each of these pairs is a copy of N#N. This manifold is null cobordant, indeed it is the boundary of N × [0, 1]. Further, as N is simply-connected, the null cobordism can be realized by a sequence of surgeries of index greater than 1.

The collection of all these surgeries is uniformly locally finite, and thus can be carried out simultaneously without leaving the bounded geometry category. Using surgery in positive scalar curvature (see [10] and [17]), we can also carry out the surgeries keeping positive scalar curvature. Note that to keep the metric in bounded distortion class of the connected sum, one changes the metric only in a neighborhood of the sphere one is surgering. The resulting distortion in that neighborhood depends on the lower bound for the scalar curvature in M, and thus the assumption of uniformly positive scalar curvature cannot be weakened to merely positive scalar curvature.

Now view M̂ similarly, but with the copies of N paired as p(2i − 1) and p(2i). One can similarly carry out surgeries here, leaving just a single copy of N at p(0). Thus, since M̂ has positive scalar curvature so does M#N. q.e.d.

Notice that we did not really need that p was a geodesic, just that the time it spends in any ball is uniformly bounded in the radius. Thus to carry out the above construction for M#_{S}N, one needs such tails from each point of S which spread out in such a way that only a uniformly bounded number pass through any ball. The existence of such “tails” is a homological question.

We say that c, an i-chain in M, is uniformly finite if there is a bound on the diameter of simplices in the support of c and for every r there is an upper bound C_{r} on the sum of the absolute values of the coefficients of the simplices which intersect any r-ball. We denote these chains by C_{i}^{uf}(M), and the corresponding homologies by H_{i}^{uf}(M).

Any S as above gives a natural element of C_{0}^{uf}(M), and a collection of tails gives a null homology of that class. We denote that class corresponding to S in H_{0}^{uf}(M) by [S]. We have sketched the proof of:

Theorem 1.2 ([6]). Let M, N, and S be as above. If [S] = 0 then M#_{S}N admits a metric of positive scalar curvature.

To get a feel for the nature of the obstruction measured by H_{0}^{uf}, consider the lattice ℤ^{2} in ℝ^{2}. For any r, the number of lattice points in a ball of radius r is about r^{2}, while the perimeter is about r. Thus, as r increases, the number of tails crossing the boundary in some bounded region must increase unboundedly. This shows that [ℤ^{2}] is nonzero.

This is essentially the only obstruction to finding tails.

Theorem 1.3 ([5], [18]). If c ∈ C_{0}^{uf}(M) then [c] = 0 if and only if there are r and C such that for any R ⊂ M one has:

|Σ_{σ∈R}c_{σ}| ≤ Cvol (∂_{r}R).

Recall that a regular sequence is a sequence R_{i} of subsets of M for which for any r,

lim_{i→∞} \frac{vol (∂_{r}R_{i})}{vol (R_{i})} = 0.

M is called amenable if and only if there is a regular sequence in M.

Theorem 1.4 ([6]). The following are equivalent:

1. M is non-amenable.

2. H_{0}^{uf}(M) = 0.

3. For all S uniformly discrete, [S] = 0.

This means, in particular, that if, in the context of Theorem 1.2, M is non-amenable then for any N and S, M#_{S}N admits a metric of positive scalar curvature. If M is a universal cover of a compact manifold then amenability is equivalent to amenability of the fundamental group.

Thus, for example, let X be a surface of higher genus cross S^{4} connected sum any N. If Â(N) ≠ 0 then X does not admit a metric of positive scalar curvature, but the universal cover does have a metric of uniformly positive scalar curvature in the bounded distortion class of the periodic metrics.

Naturally one wants to know whether [S] in H_{0}^{uf} really is an obstruction to a metric of positive scalar curvature. That it is follows from the theory of the next section.

2. The uniformly finite Â genus

How does one obstruct metrics of positive scalar curvature? In the compact case one has the Â genus. We want to generalize this to the bounded geometry setting.

According to Chern-Weil theory (see, for example, [14]), the Â class can be defined as the cohomology class of a universal polynomial, P, in the curvature tensor. On a manifold of bounded geometry, this form is bounded, and thus represents an element of l^{∞}-cohomology.

Lemma 2.1. Let M be a manifold of bounded geometry. The l^{∞}-cohomology class of the Â class is independent of choice of metric (within the given bounded distortion class).

Proof. Let {g_{t}} be a one parameter family of metrics on M for which the induced metric g_{t} + dt^{2} on M × [0, 1] has bounded curvature. Let ω be the characteristic form, given by P, on M × [0, 1]. Write ω = α_{t} +β_{t} ∧dt, where α_{t} and β_{t} are forms on M. By the assumption of bounded curvature, these forms are bounded. Since ω is closed, \frac{d}{dt}α_{t} = dβ_{t}. Thus α_{1} − α_{0} = d(∫_{0}^{1} β_{t}), and therefore α_{0} and α_{1} define the same l^{∞}-cohomology class on M. As these are the characteristic forms on M for g_{0} and g_{1}, we see that the l^{∞}-cohomology class of the characteristic form is the same for metrics connected by such one parameter families.

For any two metrics of bounded curvature on M, g_{0} and g_{1}, which are bounded distortion equivalent, Lemma 2.6 of [8] shows that the one parameter family of metric g_{t} = tg_{1} + (1 − t)g_{0} has bounded curvature, provided that the difference of Levi-Civita connections is a bounded operator (note that by the assumption of bounded distortion equivalence, bounded means the same thing for all the metrics). It is easy to see that this is the case if the identity map (M, g_{0}) to (M, g_{1}) is not only bounded distortion, but also has bounded 2-jet.

Let g_{0} and g_{1} be any two bounded distortion equivalent metrics of bounded geometry on M. By Theorem 2.5 of [8], we may assume that not only do these metrics have bounded curvature, but that their curvature tensors have bounded covariant derivatives to arbitrary order. Thus there is some r > 0 so that the exponential maps on the balls of radius r are bounded distortion diffeomorphisms with bounded derivatives of arbitrary order. The standard proof that C^{1} diffeomorphic smooth manifolds are C^{∞} diffeomorphic by convolution with a smoothing kernel (see, for example, [12], Section 2.2), shows that there is a map f : (M, g_{0}) → (M, g_{1}) at finite distance from the identity, which is a bounded distortion diffeomorphism with bounded 2-jet. Since f induces the identity on top dimensional l^{∞} cohomology, this shows that there is a well defined l^{∞}Â genus for every bounded distortion diffeomorphism class of bounded geometry metrics on M. q.e.d.

This is related to the discussion in the previous section as l^{∞}- cohomology is naturally Poincare dual to uniformly finite homology ([2]). This fact is easy to prove in the limited case we use:

Lemma 2.2. Let M^{m} be a complete, connected, Riemannian manifold of bounded geometry. There is a canonical isomorphism between H_{0}^{uf}(M) and H_{∞}^{m}(M).

Proof. Let ε > 0 be much smaller than the convexity radius of M. Let S be a maximal subset of M such that any two points of S are at distance at least ε. The balls of radius ε centered at points of S cover M, and the concentric balls of radii \frac{ε}{2} are disjoint. As S with its induced metric is quasi-isometric to M, H_{0}^{uf}(S) = H_{0}^{uf}(M).

Choose a partition of unity {f_{s}}_{s∈S} so that f_{s} is supported in B_{s}(ε). Given the bounds on the geometry of M these can be chosen with uniformly bounded derivatives. Let ϕ_{s} be the bump form \frac{f_{s}dvol}{∫ f_{s}}.

Given c ∈ C_{0}^{uf}(S), let w_{c} = Σ_{s}c_{s}ϕ_{s}. If c = ∂b for b ∈ C_{1}^{uf}(S) then b is a uniformly locally finite sum of pairs (s, s′) with d(s, s′) uniformly bounded. The difference ϕ_{s} − ϕ_{s′} is therefore d of a bounded n − 1 form of uniformly bounded support. Thus [w_{c}] = 0 in l^{∞} cohomology, so c ↦ w_{c} induces a well defined map H_{0}^{uf}(S) → H_{∞}^{n}(M).

Similarly, given w an l^{∞} n-form on M, define c_{w} = Σ_{s}(∫ f_{s}w)s. If w = dη for an l^{∞} form η, then by Theorem 1.3 and Stokes’ Theorem, c_{w} = 0 in H_{0}^{uf}(S). Thus we have a map H_{∞}^{n}(M) → H_{0}^{uf}(S). We now show these maps are inverses.

Given a uniformly finite chain c = Σc_{s}s let c′ be the image of c under the composition of these maps. We have c′ = Σ_{s}c_{s}Σ_{t}(∫ f_{t}ϕ_{s})t. Let d_{s} = Σ_{t}(∫ f_{t}ϕ_{s})t, then c − c′ = Σ_{s}c_{s}(s − d_{s}). The chain s − d_{s} is of uniformly bounded support and sum to zero, hence it is a boundary of a uniformly bounded 1-chain, b_{s} with support in a uniformly bounded neighborhood of s. Thus the difference c − c′ = ∂b = ∂Σ_{s}c_{s}b_{s} with b ∈ C_{1}^{uf}.

Likewise, given an l^{∞} form w, let w′ be the image under the composition of the two maps. We have w = Σ_{s}f_{s}w, and w′ = Σ_{s}(∫ f_{s}w)ϕ_{s}. Thus w − w′ = Σ_{s}η_{s}, where the forms η_{s} = f_{s}w − (∫ f_{s}w)ϕ_{s} are of uniformly bounded support, uniformly bounded pointwise norm, and have integral zero. This implies, by the bounded geometry of M, that η_{s} = dσ_{s} for σ_{s} also of uniformly bounded norm and support. Thus w − w′ = d(Σ_{s}σ_{s}), and therefore w = w′ in H_{∞}^{n}(M).

Thus the maps are inverses and give the desired isomorphism. q.e.d.

For the infinite connected sums we have been discussing, one has Â^{uf}(M#_{S}N) = Â^{uf}(M)+Â(N)[S], where Â(N) is the (integer) Â-genus of N. Thus the hypotheses of the results of Block and Weinberger are precisely the vanishing of the Â^{uf} genus. Our main theroem shows that the Â^{uf} genus is an obstruction to positive scalar curvature.

Theorem 2.3. If M has nonnegative scalar curvature then Â^{uf} = 0.

The proof of this theorem is, in outline, much like the corresponding theorem in the compact case: relate the Â class to the index of a Dirac operator via the index theorem and then prove the vanishing of this index by a Bochner type argument. Both steps are substantially more difficult here, and some essentially new ingredients are needed.

We first need to reinterpret Theorem 1.3 cohomologically. Cheeger and Gromov ([7]) prove a chopping theorem for manifolds of bounded geometry which says that for any n there are constants C_{0}, C_{1}, …, C_{n} and r so that for any S ⊂ M there is a codimension 0 manifold with boundary (X, ∂X) such that:

(1) S ⊂ X ⊂ N_{r}(S).

(2) For i = 0, 1, …, n ∇^{i}II_{∂X} ≤ C_{i} where II_{∂X} is the second fundamental form, and ∇^{i} is the i^{th} covariant derivative.

(3) \frac{Vol (∂X)}{Vol (∂_{r}(S))} is bounded above and below independent of S.

Lemma 2.4. For any n there are constants C_{0}, C_{1}, …, C_{n} and r so that if ω ∈ Ω^{m}(M) is bounded then ω is d of a bounded form if and only if for some C

|∫_{X} ω| ≤ Cvol (∂X)

for all (X, ∂X) compact, codimension 0 submanifold with ∇^{i}II_{∂X} ≤ C_{i} for i = 0, 1, …, n.

Proof. The condition is necessary by Stokes theorem. To see that it is sufficient we use Theorem 1.3. That lemma, formulated cohomologically, says that we need to show:

|∫_{S} ω| ≤ Cvol (∂_{r}S)

with an arbitrary S ⊂ M in place of X.

For any such S, approximate it by a manifold with boundary, X, via the chopping theorem above.

By (1) | ∫_{S} ω| and | ∫_{X} ω| differ by at most ||ω||Vol (∂_{r}S) which, by part (3) of the chopping theorem, is bounded above by Kvol (∂X) for some K which depends on ω and M, but not on S.

Thus the bound on | ∫_{S} ω| for arbitrary S follows, with perhaps a larger C, from the bound for submanifolds with bounded second fundamental forms. q.e.d.

In view of Lemma 2.4, Theorem 2.3 will follow from:

Theorem 2.5. Let (X, ∂X) be a compact spin manifold of nonnegative scalar curvature. There is a C depending only on the curvature and second fundamental form so that

|∫_{X} Â| ≤ Cvol (∂X).

We prove Theorem 2.5 by a detailed analysis of the Dirac operator on a manifold with boundary.

3. Index Theory

Let D be the canonical Dirac operator on spinors (much of this section works for an arbitrary geometric operators, but we will not need this generality).

Theorem 3.1 ([1]). For any geometric operator D, there is a characteristic form ω, a polynomial P, and n ∈ ℕ, so that for any (X, ∂X) compact we have:

index (D) = ∫_{X} w + ∫_{∂X} P (II_{∂X}, ∇II_{∂X}, …, ∇^{n}II_{∂X}) + η(∂X)

where η(N) = lim_{s→0^{+}} Σ_{λ∈Spec(D_{∂})}λ^{−s}.

We will use this to prove vanishing of ω in l^{∞}-cohomology via Lemma 2.4.

Given the bounds on ∇^{i}II_{∂X}, the middle term in Theorem 3.1 is bounded by a constant multiple of Vol (∂X).

Likewise, η is bounded linearly in Vol (∂X). This is shown for the signature operator in [8] and for a wide range of geometric operators including the Dirac operator on spinors in [15].

In view of this, and Lemma 2.4, we can interpret 3.1 as saying that the characteristic form in H_{0}^{uf} is a “uniformly finite index” of our operator.

For the Dirac operator on spinors, the form ω is the Â form. The Dirac operator is related to positive scalar curvature by:

Theorem 3.2 (Lichnerowicz formula, [14]).

D^{2} = ∇^{∗}∇ + \frac{κ}{4}

where ∇ is the canonical spinor connection, and κ the scalar curvature.

So, if s is a harmonic spinor (meaning Ds = 0) on X, we have:

0 = ⟨∇^{∗}∇s, s⟩ + \frac{κ}{4}||s||^{2}.

If X were closed, we could integrate over X, and the first term on the RHS would be ||∇s||^{2}.

Then, if κ > 0 both terms on the RHS would be ≥ 0, and therefore 0. This would mean s = 0. i.e., that there are no harmonic spinors, so that the index would be zero. Then by the Atiyah-Singer index theorem the Â genus would be 0.

It is this argument we try to extend to our setting.

When one integrates the Lichnerowicz formula over an X with boundary there is an extra term which comes from the boundary term of integration by parts (see [14]).

∫_{X}⟨∇^{∗}∇s, s⟩ = ||∇s||^{2} − ∫_{∂X}⟨∇_{ν}s, s⟩

(ν is the unit normal vector to ∂X):

The second term on the RHS is introduced when we integrate by parts.

So, when we have boundary, the Lichnerowicz formula becomes:

Lemma 3.3. If s is a harmonic spinor on X, we have:

0 = ||∇s||^{2} + ∫_{X} \frac{κ}{4}||s||^{2} − ∫_{∂X}⟨∇_{ν}s, s⟩

If we assume the scalar curvature is ≥ 0 then the first two terms are nonnegative. This can only happen if:

∫_{∂X}⟨∇_{ν}s, s⟩ ≥ 0.

We can expand D in normal coordinates around the boundary:

Lemma 3.4 ([9]). Along ∂X we have:

D = G (D_{∂X} − ∇_{ν} − \frac{1}{2}tr (II))

where G is the bundle automorphism induced by clifford multiplication by the normal vector, and D_{∂X} is the Dirac operator intrinsic to ∂_{X}.

Ds = 0 gives

⟨∇_{ν}s, s⟩ = ⟨D_{∂X}s, s⟩ − \frac{1}{2}tr (II)||s||^{2}.

Thus we have proven:

Proposition 3.5. If X has nonnegative scalar curvature, and s is a harmonic spinor, then:

∫_{∂X}⟨D_{∂X}s, s⟩ ≥ ∫_{∂X} \frac{1}{2}tr (II)||s||^{2}.

The boundary conditions for harmonic spinors in the Atiyah-Patodi-Singer index theorem are that when s|_{∂X} is expanded in eigenfunctions of D_{∂X}, only negative eigenvalues are used.

Writing s as Σ_{λ}a_{λ}s_{λ}, where the s_{λ} are the eigenvectors of D_{∂X}, 3.5 becomes:

Σ_{λ}λ||a_{λ}||^{2} ≥ \frac{1}{2} ∫_{∂X} tr (II)||s||^{2}.

Since all the λ must be negative, we must have some λ ≥ λ_{0} = inf(\frac{1}{2}tr(II)). By projecting onto these eigenspaces between λ_{0} and 0, we get:

Theorem 3.6. If (X, ∂X) is spin and has nonnegative scalar curvature then there is Λ, depending only on the second fundamental form, such that dim(H) ≤ N_{D_{∂X}^{2}} (Λ), where H is the space of harmonic spinors on X with A.-P.-S. boundary values, and N_{D_{∂X}^{2}} (Λ) is the dimension of the space of eigenfunctions of D_{∂X}^{2} below Λ.

Theorem 3.7. If N^{n} is a compact spin manifold, then for each λ there is a C_{λ} depending only on the curvature and injectivity radius of N, for which N_{D^{2}}(λ) ≤ C_{λ}vol (N).

Proof. By ([4], Prop 4.20(ii)) there are A and B so that λ_{n}(D^{2}) ≥ λ_{An−B}(Δ), where Δ is the laplacian on functions. Thus the bound we need follows immediately from the same statement (with different C_{λ}) for the laplacian on functions.

Theorem 3.8 ([11], Appendix C_{+}). There is a constant K depending only on curvature bounds and dimension for which for any compact manifold N we have the bound:

λ_{V (ε)} ≥ Kε^{−2}

for any ε ≤ inj rad. Here V (ε) is the minimal number of ε balls which cover N.

Since we have bounds on the curvature, there is an L such that:

Vol (B(ε)) ≥ Lε^{n}

for B(ε) any such ε ball in N.

Choose a maximal family of disjoint \frac{ε}{2} ball in N. By maximality the concentric balls of radius ε cover. But by disjointness there are at most

\frac{2^{n}Vol (N)ε^{−n}}{L}

balls. Therefore we have:

Proposition 3.9. There is a constant C depending only on the curvature such that, for any ε ≤ inj rad(N)

λ_{Cvol (N)ε^{−n}} ≥ ϵ^{−2}.

Turning this around to an upper bound on the spectral counting function gives:

Proposition 3.10. There is a constant C depending only on the curvature, and a λ_{0} depending only on curvature and injectivity radius, such that for any λ ≥ λ_{0}

N(λ) ≤ CVol (N)λ^{\frac{n}{2}} .

This gives the C_{λ} we needed for 3.7 C_{λ} = Cλ^{\frac{n}{2}} works for λ ≥ λ_{0} and C_{λ} = Cλ0^{\frac{n}{2}} works for λ ≤ λ_{0}, as N(λ) ≤ N(λ_{0}). q.e.d.

This completes the proof of Theorem 2.5, and thereby Theorem 2.3.

4. Applications

Our first corollary, combined with the work of [6], gives a complete characterization of infinite connected sums of positive scalar curvature.

Corollary 4.1. Let M, N, and S be as before. If Â(N) ≠ 0 then M#_{S}N admits a metric of uniformly positive scalar curvature if and only if [S] = 0 in H_{0}^{uf}(M).

In fact, 2.3 shows that if [S] ≠ 0 then M#_{S}N does not even admit a metric of nonnegative scalar curvature. Thus there is an interesting alternative: an infinite connected sum where M has uniformly positive scalar curvature either has a metric of uniformly positive scalar curvature or does not even have a metric of nonnegative scalar curvature. It would be interesting to understand what happens when M only has nonnegative scalar curvature. Theorem 2.3 still gives [S] as an obstruction, but is very likely no longer sharp. As noted in the sketch of the proof of Theorem 1.2, the construction there does not give metrics of bounded geometry unless one has a positive lower bound on κ. It seems likely that in the place of l^{∞}-cohomology one needs forms which go to zero at ∞ in some way related to κ. A closer examination of the behavior of C in Theorem 2.5 in terms of the lower bound on κ might give the right decay condition.

One intriguing aspect of Theorem 2.3 is that the obstruction lives in a non-Hausdorff group. This prevents the problem from fitting in to the C^{∗} algebra framework usually used for these types of problems. To get around this, one can work with reduced uniformly finite homology, where the groups are the quotients of cycles by the closure of the boundaries. The reduced invariant is shown to obstruct positive scalar curvature in [16] and [6]. Corollary 4.1 shows that this loses important information. The necessity of working with non-Hausdorff groups makes it unclear how to get versions of the theorems for families, and in particular, it is unclear when (M#_{S}N) × ℝ^{i} carries a metric of positive scalar curvature.

Theorem 2.5 has applications to compact manifolds as well. Recall the theorem of [13] which says that for any compact manifold there is a metric whose non-positive scalar curvature is contained in an arbitrary ball. This is a purely topological statement. As a corollary of 2.5, one can see that this ball cannot be arbitrarily small.

Theorem 4.2. Let M be a compact spin manifold with Â ≠ 0. For any bounds on the curvature, there is an r > 0 so that there is no metric on M with those bounds on curvature and whose non-positive curvature set is contained in a ball of radius r.

Proof. For r small enough the ball is embedded. Further, one has bounds on the second fundamental form of the sphere in terms of r and the curvature of M. Applying Theorem 2.5 to the complement of the ball gives a bound, which goes to 0 with r, on the integral of the Â class over the complement of the ball. As the curvature is bounded, the integral over the ball is bounded by a multiple of its volume. These must add to a nonzero integer, which is a contradiction for r sufficiently small. q.e.d.

Also, many of the standard applications (see, for example, [14]) of the Bochner method can be carried over to manifolds with boundary as well: flat manifolds have their signature bounded by a multiple of the volume of the boundary, likewise for the total Betti number of manifolds with positive definite curvature tensor. For submanifolds of Euclidean space or the sphere, these results follow easily just from Alexander duality.

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Dept. of Mathematics University of Chicago