PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0

POSITIVE SCALAR CURVATURE FOR MANIFOLDS WITH ELEMENTARY ABELIAN FUNDAMENTAL GROUP

BORIS BOTVINNIK AND JONATHAN ROSENBERG

(Communicated by Wolfgang Ziller)

Abstract. The statement often called the Gromov-Lawson-Rosenberg Conjecture asserts that a manifold with finite fundamental group should admit a metric of positive scalar curvature except when the KO_{∗}-valued index of some Dirac operator with coefficients in a flat bundle is non-zero. We prove spin and oriented non-spin versions of this statement for manifolds (of dimension ≥ 5) with elementary abelian fundamental groups π, except for “toral” classes, and thus our results are automatically applicable once the dimension of the manifold exceeds the rank of π. The proofs involve the detailed structure of BP_{∗}(Bπ), as computed by Johnson and Wilson.

1. Introduction

A fundamental and fascinating problem in global differential geometry is to determine necessary and sufficient conditions for a closed manifold to admit a Riemannian metric with scalar curvature function everywhere positive. For surveys on this question, see [17] and [15].

In this paper we consider this problem only for manifolds (of dimension at least five) with finite fundamental group. By a result of Kwasik and Schultz [9], the problem is then further reduced to the case where the fundamental group is a finite p-group. Somewhat different techniques are needed depending on whether or not the universal cover of the given manifold admits a spin structure. Here we will consider only the following two important cases: when the manifold admits a spin structure, and when it is oriented and its universal cover is not spin. (If p is odd, these are the only two cases.)

When the universal cover is non-spin, there are no known obstructions to positive scalar curvature. In the spin case, the only known obstructions to positive scalar curvature come from the index theory of the Dirac operator ([14], [15]). In fact, in [12], it was conjectured (on the basis of extremely spotty evidence) that the Dirac operator index obstructions are the only obstructions to positive scalar curvature on manifolds of dimension ≥ 5 with finite fundamental group. This conjecture has sometimes been called the Gromov-Lawson-Rosenberg Conjecture.

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

1991 Mathematics Subject Classification. Primary 53C20. Secondary 53C21, 55S30, 55N22, 55U25, 57R75.

We thank Sergey Novikov for helping to make this collaboration possible.

Partially supported by NSF grant DMS-0103647.

It is known that “stably” these are the only obstructions [14]. This means that for such a manifold M there exists an integer k ≥ 0 such that the product M ×(J^{8})^{k} admits a positive scalar curvature metric if and only if the Dirac obstructions vanish. Here J^{8} is a simply connected spin manifold with Â(J^{8}) = 1. Such manifold J^{8} does not admit positive scalar curvature, and geometrically represents Bott periodicity in KO-theory. Moreover, J^{8} can be chosen to be a Ricci-flat “Joyce manifold” [8] with holonomy Spin(7). The “stable” theorem by itself does not actually answer the question of whether any particular manifold with vanishing Dirac obstructions admits a metric of positive scalar curvature.

Botvinnik, Gilkey and Stolz [4] proved the Gromov-Lawson-Rosenberg Conjecture for spin manifolds with finite fundamental group with periodic cohomology. A similar theorem was proved by Schultz [16], and independently by Botvinnik and Gilkey [3], for spin manifolds of dimension ≥ 5 with fundamental group ℤ/p × ℤ/p, p an odd prime. But very little was previously known about the existence of positive scalar curvature for manifolds with elementary abelian fundamental group π = (ℤ/p)^{r} of rank r > 2. That is the subject of this paper. We build on some results of our earlier paper, [5]. Our main results are Theorems 2.3 and 2.4.

2. Main Results

Let π = (ℤ/p)^{r} be an elementary abelian p-group, where we call r the rank of π. We will be interested in the Gromov-Lawson-Rosenberg Conjecture for manifolds M with fundamental group π. We will deal with the following cases:

(1) p odd. The sub-case where M is non-spin was already treated in [5], so we will concentrate here on the sub-case where M is spin, though the same proof will work for the non-spin case, also.

(2) p = 2, M oriented, and the universal cover M̃ is non-spin.

Recall that since any oriented (resp., spin) manifold is automatically oriented for ordinary homology (resp., connective or periodic real K-theory) that there are natural transformations Ω_{∗}( ·) → H_{∗}( ·) and Ω_{∗}^{spin}( ·) → ko_{∗}( ·). The composite

Ω_{∗}^{spin}(X) → ko_{∗}(X) → KO_{∗}(X) → KO_{∗}(point)

corresponds to taking the generalized index of the Dirac operator, and is traditionally denoted α.

To state our main results, we have to single out certain bordism classes.

Definition 2.1. If k ≤ r and one fixes a homomorphism ℤ^{k} ↠ (ℤ/p)^{k} ↪ (ℤ/p)^{r}, taking classifying spaces gives a map T^{k} = B(ℤ^{k}) → B(ℤ/p)^{r}. The collection of all such maps is said to generate the subgroup of toral bordism classes in Ω_{k}(Bπ) or in Ω_{k}^{spin}(Bπ). We define a complement to the toral classes in the following way. Consider the natural Thom map Ω_{k}(Bπ) → H_{k}(Bπ, 𝔽_{p}) or Ω_{k}^{spin}(Bπ) → H_{k}(Bπ, 𝔽_{p}), where 𝔽_{p} is the field of p elements. When p is odd, we have a natural cup product map ⋀^{k} H^{1}(Bπ, 𝔽_{p}) ↪ H^{k}(Bπ, 𝔽_{p}), whose image (the group of toral

cohomology classes) has dimension (\begin{matrix}r \\ k \end{matrix}). And even when p = 2, even though in

this case the classes in H^{1}(Bπ, 𝔽_{p}) do not have square 0, we still have a similar subgroup of H^{k}(Bπ, 𝔽_{p}), spanned by products α_{i_{1}} ⋯ α_{i_{k}}, 1 ≤ i_{1} < ⋯ < i_{k} ≤ r, where α_{1}, ⋯, α_{r} is a basis for H^{1}(Bπ, 𝔽_{p}). (The subgroup one gets in this case depends on the choice of basis.) A bordism class is called atoral if its image in H_{k}(Bπ, 𝔽_{p}) is annihilated by these toral cohomology classes. Toral and atoral classes in H_{∗}(Bπ) or ko_{∗}(Bπ) are defined to be the images of toral or atoral classes in bordism or spin bordism. Since the pairing between the toral homology and cohomology classes is non-degenerate, and since the non-zero toral bordism classes are all of order p (since the bordism class of S^{1} → Bℤ/p is of order p), the group of toral bordism classes is an elementary abelian p-group, and Ω_{k}(Bπ) or Ω_{k}^{spin}(Bπ) splits as the direct sum of this group and the group of atoral classes.

Remark 2.2. (a) It is worth remarking, since we will need this observation later, that if M^{k} is an oriented or spin k-manifold whose fundamental group can be generated by at most k − 1 elements, then for any map f: M → Bπ, the bordism class of M ⟶^{f} Bπ is atoral.

(b) Another useful fact which follows immediately from the definition is that the group of toral classes (and when p is odd, also the group of atoral classes) is preserved under injective group homomorphisms from one elementary abelian p-group to another.

Now we can state our main results:

Theorem 2.3. Let p be an odd prime, let π be an elementary abelian p-group, and let n ≥ 5. Let M be a spin manifold with α(M) = 0 in KO_{n}, and let f: M → Bπ be the classifying map for its universal covering. If the class [M ⟶^{f} Bπ] ∈ Ω_{n}^{spin}(Bπ) is atoral, then M has a metric with positive scalar curvature. In particular, if n > rank π, then every spin n-manifold with vanishing α-invariant and fundamental group π has a metric of positive scalar curvature. (The same result holds for nonspin manifolds, with Ω^{spin} replaced by Ω and without the condition on α(M).)

Theorem 2.4. Let π be an elementary abelian 2-group, and let n ≥ 5. Suppose M^{n} is a closed oriented manifold with fundamental group π and non-spin universal cover, and let f: M → Bπ be the classifying map for its universal covering. If the class [M ⟶^{f} Bπ] ∈ H_{n}(Bπ, ℤ) is atoral, then M has a metric with positive scalar curvature. In particular, if n > rank π, then every oriented n-manifold with fundamental group π and with non-spin universal cover has a metric of positive scalar curvature.

Remark 2.5. The case when p is odd and M is non-spin was already treated in [5]. For p = 2, the case when M is spin is very different from the non-spin case since Ω_{∗}^{spin} (B(ℤ/2)^{r}) has a rather different structure from Ω_{∗}^{SO} (B(ℤ/2)^{r}), and also because in the spin case there are many Dirac obstructions to positive scalar curvature.

3. Topological Preliminaries

Our basic tool will be the following Bordism/Homology Theorem, due to a combination of work of Gromov-Lawson, Rosenberg-Stolz, Stolz, and Jung, which can be found in [14] and [15].

Theorem 3.1 (Bordism/Homology Theorem). Let M^{n} be a (connected, closed) oriented manifold with n = dim M ≥ 5, and let f: M → Bπ be the classifying map for its universal covering. (In other words, Bπ is a K(π, 1)-space and f induces an isomorphism on π_{1}. Existence and essential uniqueness of f is guaranteed by obstruction theory.)

(1) If M is spin, M admits a metric of positive scalar curvature if and only if there is some spin manifold of positive scalar curvature in the same spin bordism class as M ⟶^{f} Bπ, in fact if there is some spin manifold of positive scalar curvature representing the same class in ko_{n}(Bπ) as M ⟶^{f} Bπ.

(2) If the universal cover M̃ of M is non-spin (i.e., w_{2}(M̃) ≠ 0), M admits a metric of positive scalar curvature if and only if there is some oriented manifold of positive scalar curvature in the same oriented bordism class as M ⟶^{f} Bπ, in fact if there is some oriented manifold of positive scalar curvature representing the same class in H_{n}(Bπ, ℤ) as M ⟶^{f} Bπ.

Before proceeding further, we would like to remind the reader of some known results on Ω_{∗}(Bπ) and Ω_{∗}^{spin}(Bπ), where π is an elementary abelian p-group of rank r with p an odd prime. Recall that

(Ω_{∗})_{(p)} ≅ ℤ_{(p)}[x_{4}, x_{8}, …, x_{4k}, …]

with |x_{4k}| = 4k. Let P = ℤ_{(p)}[ x_{4i} | 2i ≠ p^{ℓ} −1 ], and M(P) denote the corresponding generalized Moore spectrum (just a wedge of shifted sphere spectra, localized at p). Then there is a splitting of the spectrum

(3.1) MSO_{(p)} = BP ∧ M(P) ≅ ⋁_{i_{1},j_{1},i_{2},j_{2},···≥0} Σ^{4i_{1}^{j_{1}} +4i_{2}^{j_{2}} +···}BP,

where 2i_{1}, 2i_{2}, ⋯ ≠ p^{ℓ} − 1.

Here BP_{∗} = ℤ_{(p)}[v_{1}, …, v_{k}, …], |v_{j}| = 2(p^{j} − 1). In particular, we have a BP_{∗}- module isomorphism (Ω_{∗})_{(p)} ≅ BP_{∗} ⊗ P. Our goal is to describe the image of the map Ω_{∗}(Bπ) → H_{∗}(Bπ; ℤ) or Ω_{∗}^{spin}(Bπ) → ko_{∗}(Bπ). But since π is an elementary abelian p-group with p odd, it is enough to work in the p-local category; furthermore, the natural map MSO → MSpin is a p-local equivalence, so Ω̃_{∗}^{spin}(Bπ) ≅ Ω̃_{∗}(Bπ). Furthermore, (ko_{∗})_{(p)} ≅ ℤ_{(p)}[ω_{4}] for p odd, and the generators x_{4i} and v_{j} may be chosen [3, Proposition 2.1] so that the map of rings Ω_{∗}^{spin} → ko_{∗} corresponds to killing all the v_{j}, j > 1, and x_{4i}, i > 4, and sending either v_{1} (when p = 3) or x_{4} (when p > 3) to ω_{4}. (When p > 3, then v_{1} ↦ cω_{4}^{(p−1)/2} for some c ≠ 0.) The map Ω̃_{∗}(Bπ) → H̃_{∗}(Bπ; ℤ) always factors through B̃P_{∗}(Bπ), and so does the map Ω̃_{∗}^{spin}(Bπ) → k̃o_{∗}(Bπ) when p = 3. (When p > 3, it factors through a wedge of p−2 copies of B̃P_{∗}(Bπ).) So it is enough to work with BP_{∗}(Bπ). We need the following results on the structure of BP_{∗}(Bℤ/p); see, say, [7]. We have that BP^{∗}(ℂℙ^{∞}) = BP^{∗}[[x]], where x ∈ BP^{2}(ℂℙ^{∞}) is the (BP version of the) first Chern class. Here we identify BP^{−∗} = BP_{∗}. Start with the standard fibration

Bℤ/p⟶ℂℙ^{∞} ⟶^{p} ℂℙ^{∞},

where the map p: ℂℙ^{∞} = K(ℤ, 2) → K(ℤ, 2) = ℂℙ^{∞} induces multiplication by p on π_{2}(ℂℙ^{∞}). Then p^{∗} : BP^{∗}(ℂℙ^{∞}) → BP^{∗}(ℂℙ^{∞}) is given as

p^{∗}(x) = [p](x) = ∑_{i≥0} a_{i}x^{2i+1}, a_{i} ∈ BP_{4i}.

(We have re-indexed from the way things are written in [7], since if p is odd, then BP_{∗} is concentrated in degrees divisible by 4.) In particular, BP^{∗}(Bℤ/p) ≅ BP^{∗}[[x]]/[p](x). Let I_{n} = (p, v_{1}, …, v_{n−1}). The coefficients a_{i} satisfy

• a_{i} ∈ I_{n} for i < (p^{n} − 1)/2;

• a_{(p^{n}−1)/2} ≡ v_{n} mod I_{n}, in particular, a_{0} = p, and a_{i} = 0 for 0 < i <

(p − 1)/2, and a_{(p−1)/2} = v_{1}.

The BP_{∗}-module N_{∗} = B̃P_{∗}(Bℤ/p) is generated by elements

z_{2m−1} ∈ BP_{2m−1}(Bℤ/p)

(represented by the standard lens spaces), subject to the relations:

(3.2)  R_{m} = ∑_{i=0}^{m−1} a_{i}z_{m−4i} = 0.

In particular, N_{∗} has homological dimension 1 as a BP_{∗}-module. It was shown in [10] that B̃P_{∗} (B(ℤ/p)^{r}) may be computed via iterated applications of a Künneth formula involving · ⊗_{BP_{∗}} N_{∗} and Σ Tor_{BP_{∗}}( · , N_{∗}). Let L_{k} be the free BP_{∗}-module on generators of degree 2k, 1 ≤ k ≤ p^{k} − 1, and let π = (ℤ/p)^{r}. It is shown in [7] that Σ Tor_{BP_{∗}}(N_{∗}, N_{∗}^{⊗^{j}}) ≅ L_{j} ⊗ N_{∗}^{⊗^{j}}, and thus that there is a natural filtration on B̃P_{∗}(Bπ), coming from the Landweber Künneth Theorem, for which the associated graded BP_{∗}-module is a direct sum of tensor powers N_{∗}^{⊗^{j}}, 1 ≤ j ≤ r (these correspond to pure product terms in the Künneth formula) and tensor products of tensor powers of N with tensor powers of L_{1}, ⋯, L_{r−1}, where these terms have geometrical generators involving Toda brackets.

For future use, we also record a few more facts about the natural transformations BP_{∗} → H_{∗} or ⋁_{j=1}^{p−2} Σ^{4j}BP_{∗} → ko_{∗}. Via these maps, the Eilenberg-Mac Lane spectrum H = Hℤ_{(p)} becomes a BP -module spectrum, so for (p-local) spaces X with BP_{∗}(X) of finite homological dimension over BP_{∗} there is a convergent universal coefficient spectral sequence [1, §1, (UCT1)]

(3.3)  Tor_{BP_{∗}}^{s}(BP_{t}(X), ℤ) ⇒ H_{s+t}(X, ℤ)

for which one of the edge homomorphisms is

E_{0,n}^{2} = BP_{n}(X) ⊗_{BP_{∗}} ℤ → H_{n}(X, ℤ).

The map BP_{n}(X) → H_{n}(X, ℤ) factors through this map (which simply kills v_{1}, v_{2}, ⋯). Similarly (again by [1, §1]) there is a universal coefficient spectral sequence

(3.4)  Tor_{BP_{∗}[x_{4}]}^{∗}(BP_{∗}(X)[x_{4}], ko_{∗}) ⇒ ko_{∗}(X)

for which one of the edge homomorphisms is

BP_{∗}(X)[x_{4}] ⊗_{BP_{∗}} ℤ[ω_{4}] → ko_{∗}(X).

Notation 3.2. As in [3], we write 𝔨𝔬_{∗}(X) for BP_{∗}(X)[x_{4}] ⊗_{BP_{∗}} ℤ[ω_{4}], the result of killing v_{2}, v_{3}, ⋯ and mapping x_{4} ↦ ω_{4}, v_{1} ↦ cω_{4}^{(p−1)/2} for suitable c ≠ 0. The Thom maps BP_{n}(X) → H_{n}(X, ℤ) and BP_{n}(X) → ko_{n}(X) thus factor through 𝔨𝔬_{n}(X), though in general the map 𝔨𝔬_{∗}(X) → ko_{∗}(X) is neither injective nor surjective. However, one can check (from the BP_{∗}-resolution of N_{∗} with relations (3.2)) that for X = Bℤ/p, (3.4) collapses and gives an isomorphism 𝔨𝔬_{∗}(Bℤ/p) ≅ ko_{∗}(Bℤ/p). In particular, the map Ω_{∗}^{spin}(Bℤ/p) → ko_{∗}(Bℤ/p) is surjective.

4. Proofs of the Main Results for p Odd

The following fact will be our key technical tool in the proof of Theorem 2.3. The homotopy functor 𝔨𝔬( ·) (not a homology theory) is defined in Notation 3.2.

Proposition 4.1. Let π be an elementary abelian p-group of rank 2, where p is an odd prime. Then 𝔨𝔬_{odd}(Bπ) is generated (as an abelian group) by the images of 𝔨̃𝔬_{∗}(Bσ), as σ runs over the cyclic subgroups of π.

Proof. This is proved in [3], using explicit calculations of the eta-invariants of lens spaces. □

Now we are ready to prove the following result.

Theorem 4.2. Let π be an elementary abelian p-group of rank r, where p is an odd prime. Then the image of Ω_{∗}^{spin}(Bπ) in ko_{∗}(Bπ) is generated (as an abelian group) by images of elements x_{1} ⊗ ⋯ ⊗ x_{j} ∈ Ω_{∗}^{spin}(Bσ_{1}) ⊗ ⋯ ⊗ Ω_{∗}^{spin}(Bσ_{j}), with σ_{1} × ⋯ × σ_{j} ↪ π, j ≤ r, and with each σ_{i} a cyclic p-group. (However, the embedding σ_{1} × ⋯ × σ_{j} ↪ π is not necessarily the “standard” one.)

Proof. We prove this by induction on the rank r. When r = 1, the statement is trivially true, and when r = 2, this is Proposition 4.1. Now assume the result for smaller values of r, and write π = π′ × ℤ/p, where π′ has rank r − 1. We use the BP -version of the diagram (4.1) in [5, Proof of Theorem 4.1]:

(4.1) [figure: 0; BP_{∗}(Bπ′)⊗_{BP∗} BP_{∗}(Bℤ/p); α; BP_{∗}(Bπ); β; Tor_{BP∗}^{1}(BP_{∗}(Bπ′), BP_{∗}(Bℤ/p)); γ; 0; 𝔨𝔬_{∗}(Bπ′)⊗_{ko∗} ko_{∗}(Bℤ/p); 𝔨𝔬_{∗}(Bπ); Tor_{ko∗}(𝔨𝔬_{∗}(Bπ′), ko_{∗}(Bℤ/p)); 0; ko_{∗}(Bπ′)⊗_{ko∗} ko_{∗}(Bℤ/p); ko_{∗}(Bπ); Tor_{ko∗}(ko_{∗}(Bπ′), ko_{∗}(Bℤ/p)); 0.]

Here the bottom row comes from the Künneth Theorem in ko_{∗}, which applies to this case (see again [1]) since Bπ′, Bℤ/p, and Bπ are p-local for p odd. In the top row, if p > 3 one should really replace BP_{∗} by BP [ω_{4}]_{∗} (but we’ve left off the [ω_{4}] to save space). The image of α is taken care of by inductive hypothesis. Also by inductive hypothesis, the image of γ is generated by the images of

(4.2) Tor_{BP [ω]_{∗}}^{1}(BP [ω]_{∗}(Bσ_{1}) ⊗_{BP [ω]_{∗}} ⋯ ⊗_{BP [ω]_{∗}} BP [ω]_{∗}(Bσ_{j}), BP [ω]_{∗}(Bℤ/p))

for the various subgroups σ_{1} × ⋯ σ_{j} ↪ π′. The image of (4.2) is contained in a copy of the image of BP [ω]_{∗} (B(ℤ/p)^{r−1}) if j < r − 1, which is also covered by the inductive hypothesis. So we may assume j = r − 1. Thus we are reduced to studying the image of the map (4.3)

γ: Tor_{BP_{∗}}^{1}(\overbrace{N_{∗} ⊗_{BP_{∗}} ⋯ ⊗_{BP_{∗}} N_{∗}}^{r−1}, N_{∗})[ω] → Tor_{ko_{∗}}(\overbrace{P_{∗} ⊗_{ko_{∗}} ⋯ ⊗_{ko_{∗}} P_{∗}}^{r−1}, P_{∗}),

where P_{∗} = k̃o_{∗}(Bℤ/p) as a module over ko_{∗} = ℤ[ω]. (Recall we have localized at p. Also recall from Section 3 that P_{∗} = N_{∗}[ω] ⊗_{BP [ω]_{∗}} ko_{∗}.) The BP_{∗}-module on the left in (4.3) is computed in [7, Theorem 4.1]; this subquotient of BP_{∗}(Bπ) is a direct sum

Tor_{BP_{∗}}^{1}((N_{∗})^{⊗^{r−1}}, N_{∗}) ≅ ⨁_{j=1}^{p^{r−1}−1}(N_{∗})^{⊗^{r−1}}y_{2j}

of copies of (N_{∗})^{⊗^{r−1}}, shifted up in degrees by 2j, 0 < j < p^{r−1}. We need to show that the image of this graded group under (4.3) is generated by products of lens spaces.

Let I = (i_{1}, …, i_{r−1}) be a multi-index with all the i_{k}’s odd, and let z_{I} = z_{i_{1}} ⊗ ⋯ ⊗ z_{i_{r−1}} be the corresponding product of lens spaces (or the element of

(N_{∗})^{⊗^{r−1}} or (P_{∗})^{⊗^{r−1}} represented by this product). Then we need to show that the image in ko_{∗} of z_{I}y_{2j} ∈ BP_{∗}(Bπ) is represented by linear combinations of products of lens spaces, for 0 < j < p^{r−1}. Since the BP_{∗}-module N_{∗} is given in terms of the generators z_{m} and the relations R_{m}, the element z_{I}y_{2j} is represented by a linear combination of “matrix Toda brackets” [2, §5.10]

(4.4) ⟨z_{i_{1}} ⊗ ⋯ ⊗ z_{i_{k−1}} ⊗ (z_{i_{k}}, z_{i_{k}−4}, ⋯) ⊗ z_{i_{k+1}} ⊗ ⋯ ⊗ z_{i_{r−1}}, A, ( ⋮ ; z_{2j−5} ; z_{2j−1})⟩ ,

possibly with smaller values of i_{k}, where A is the matrix

(\begin{matrix}p & 0 & 0 & 0 & ⋯ \\ a_{1} & p & 0 & 0 & ⋯ \\ a_{2} & a_{1} & p & 0 & ⋯ \\ a_{3} & a_{2} & a_{1} & p & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix})

encoding the relations (3.2). By the product property of Toda brackets [2, 2.1, axiom 3], (4.4) can be rewritten as

±z_{i_{1}} ⊗ ⋯ ⊗ z_{i_{k−1}} ⊗ ⟨⋯ ⟩ ⊗ z_{i_{k+1}} ⊗ ⋯ ⊗ z_{i_{r−1}},

where the bracket in the middle corresponds to something of rank 2 to which we can apply Proposition 4.1. This gives the inductive step. □

Proof of Theorem 2.3. Now we can give the proof of the first main theorem. Recall that the atoral part of Ω_{∗}^{spin}(Bπ) was defined in Definition 2.1. By Theorem 4.2, it is enough to consider product classes

(4.5)  (M_{1}^{i_{1}} → Bℤ/p) × ⋯ × (M_{k}^{i_{k}} → Bℤ/p)

in Ω_{i_{1}+···+i_{k}}^{spin}(B(ℤ/p)^{k}), as the image of Ω_{∗}^{spin}(Bπ) in ko_{∗}(Bπ) is generated by the images of such classes under the maps Ω_{∗}^{spin}(B(ℤ/p)^{k}) → Ω_{∗}^{spin}(Bπ) induced by embeddings (ℤ/p)^{k} ↪ π, k ≤ rank π = r. But by Remark 2.2(a), the class (4.5) is toral exactly when i_{1} = ⋯ = i_{k} = 1, and atoral exactly when some i_{j} > 1, and by Remark 2.2(b), toral or atoral classes in Ω_{∗}^{spin}(B(ℤ/p)^{k}) map to toral or atoral classes in Ω_{∗}^{spin}(Bπ). So the result follows from the fact that all atoral classes in Ω̃_{∗}^{spin}(Bℤ/p) are represented by manifolds of positive scalar curvature. □

5. Proofs of the Main Results for p = 2

The parallel to Theorem 4.2 when p = 2 is the following:

Theorem 5.1. Let π be an elementary abelian 2-group. Then for any n > 1, H_{n}^{atoral}(Bπ, ℤ) is generated (as an abelian group) by classes of maps f: M^{n} → Bπ, where M is an oriented n-manifold of positive scalar curvature, and where M admits an orientation-reversing isometry c: M → M, admitting a fixed point, commuting up to homotopy with f.

Proof. Since π is an elementary abelian 2-group and MSO localized at 2 is an Eilenberg-Mac Lane spectrum (see [11] and [18], or [13, pp. 54–57] for a review of the literature) the map Ω_{∗}(Bπ) → H_{∗}(Bπ, ℤ) is split surjective. Also, we know H̃_{∗}(Bπ, ℤ) embeds in H_{∗}(Bπ, 𝔽_{2}), which pairs non-degenerately with

H^{∗}(Bπ, 𝔽_{2}) = 𝔽_{2}[x_{1}, ⋯, x_{r}],

where x_{1}, ⋯, x_{r} are cohomology generators each of degree 1. So we will detect homology classes by means of pairing with this polynomial ring. By [5, Proposition 5.2], for n ≥ 1,

dim_{𝔽_{2}} H_{n}(Bπ, ℤ) = ∑_{j=1}^{n}(−1)^{n−j} (\begin{matrix}j + r − 1 \\ r − 1 \end{matrix}) ,

while the toral subgroup has dimension (\begin{matrix}r \\ n \end{matrix}) if n ≤ r, 0 if n > r. We will prove

the result by induction on r = rank π, with the condition regarding an orientationreversing isometry needed for the inductive step.^{1}

To start the induction, take r = 1, π = ℤ/2, and Bπ = ℝℙ^{∞}. Note that H_{n}(Bℤ/2, ℤ) vanishes for n > 0 even and is generated for n ≥ 1 by the class of ℝℙ^{n} ↪ ℝℙ^{∞}. This class is atoral when n ≥ 3, exactly when it admits a metric of positive scalar curvature. Furthermore, ℝℙ^{n} (n odd) with its standard metric admits many orientation-reversing isometries, even of period 2 (for instance,

[t_{0}, t_{1}, t_{2}, ⋯, t_{n}] ↦ [t_{1}, t_{0}, t_{2}, ⋯, t_{n}]

in homogeneous coordinates), all of which preserve the unique non-trivial element of H^{1}(ℝℙ^{n}, ℤ/2) and thus commute up to homotopy with the map to ℝℙ^{∞}. Any such isometry has a fixed point (by the Lefschetz Fixed Point Theorem). So this starts the induction.

Now assume the theorem is true for smaller values of r, and write π = π′ × ℤ/2, where rank π′ = r − 1. By the Künneth Theorem, we have short exact sequence

(5.1) 0 → H_{n}(Bπ′, ℤ) ⊕ ⨁_{j=1}^{[(n+1)/2]} H_{n−2j+1}(Bπ′, ℤ) ⊗_{ℤ} H_{2j−1}(Bℤ/2, ℤ) → H_{n}(Bπ, ℤ)

→ ⨁_{j=1}^{[n/2]} Tor_{ℤ}(H_{n−2j}(Bπ′, ℤ), H_{2j−1}(Bℤ/2, ℤ)) → 0,

1Actually it would be enough for our purposes to have an orientation-reversing diffeomorphism c such that the metric and its pull-back under g lie in the same connected component of the space of metrics of positive scalar curvature. But this condition, while weaker, is more burdensome to check.

which splits (unnaturally). The “tensor” terms on the left in (5.1) are clearly no problem, as they are represented by classes of maps M^{n} → Bπ′ ×pt ↪ Bπ′ ×Bℤ/2

or M^{n−2j+1} × ℝℙ^{2j−1} ⟶^{f×ι} Bπ′ × Bℤ/2, where ι: ℝℙ^{2j−1} ↪ ℝℙ^{∞}. By inductive

hypothesis, H_{∗}^{atoral}(Bπ′, ℤ) is represented by manifolds of positive scalar curvature, hence so is H_{∗}^{atoral}(Bπ′, ℤ) ⊗ H_{∗}(Bℤ/2, ℤ). Similarly anything in H_{∗}(Bπ′, ℤ) ⊗ H_{2j−1}(Bℤ/2, ℤ) admits a metric of positive scalar curvature if j > 1. On the other hand, manifolds of the form M^{n} or M^{n−1} × S^{1}, with M toral, are themselves toral. Furthermore, the required orientation-reversing isometry on M^{n−2j+1}×ℝℙ^{2j−1} can be constructed from one on either M or ℝℙ^{2j−1}.

So that leaves only the Tor terms H_{n−2j}(Bπ′, ℤ)∗H_{2j−1}(Bℤ/2, ℤ) on the right in (5.1) to worry about. These terms are represented (non-canonically) by homology Toda brackets ⟨[M^{n−2j}], 2, z_{2j−1}⟩. We will distinguish two cases: (a) the case where either j > 1 or the class [M^{n−2j}] ∈ H_{n−2j}(Bπ′) is atoral, and (b), the exceptional case of H_{n−2}^{toral}(Bπ′, ℤ) ∗ H_{1}(Bℤ/2, ℤ). First consider the case (a). We shall show that representatives for these terms can be constructed as twisted products, i.e., fiber bundles, of the form

(5.2)  M^{n−2j} × ℝℙ^{2j−1} → N^{n} → S^{1},

where g: M → Bπ′ represents a class in H_{n−2j}(Bπ′, ℤ). By assumption, M admits an orientation-reversing isometry c_{1} with a fixed point, and similarly ℝℙ^{2j−1} admits an orientation-reversing isometry c_{2} with a fixed point. Let c = c_{1} × c_{2}; this is an isometry of M^{n−2j} × ℝℙ^{2j−1} that preserves orientation. Thus if N denotes the mapping torus of c, N is an oriented manifold fitting into a fiber sequence (5.2). Also note that N contains within it two submanifolds N_{1}^{n−2j+1} and N_{2}^{2j}, the mapping tori of c_{1} and c_{2}, respectively. Since c_{1} and c_{2} reverse orientation, N_{1} and N_{2} are not orientable. Now consider the cohomology ring H^{∗}(N, 𝔽_{2}). This ring has a filtration (associated to the Leray-Serre spectral sequence

H^{∗}(S^{1}, H^{∗}(M × ℝℙ^{2j−1}, 𝔽_{2})) ⇒ H^{∗}(N, 𝔽_{2}),

which collapses since S^{1} is 1-dimensional) for which the associated graded ring is just the tensor product

H^{∗}(M × ℝℙ^{2j−1}, 𝔽_{2}) ⊗ H^{∗}(S^{1}, 𝔽_{2}) ≅ H^{∗}(M, 𝔽_{2})[v, w]/(v^{2j}, w^{2}).

However, it will be important to note that the actual product structure on H^{∗}(N, 𝔽_{2}) is different. To prove this, consider for instance the inclusion ι: N_{2} ↪ N. (N_{2} is the mapping torus of the restriction of c to {pt} × ℝℙ^{2j−1}, with the base point in M chosen to be a fixed point of c_{1}. The case of N_{1} ↪ N is similar, except that our knowledge of the cohomology ring of M, and thus of the cohomology ring of N_{1}, is less explicit.) Note that H^{∗}(N_{2}, 𝔽_{2}) is generated by two elements of degree 1, say v and w. The first of these restricts to a generator of H^{∗}(ℝℙ^{2j−1}, 𝔽_{2}), and the second restricts to a generator of H^{∗}(S^{1}, 𝔽_{2}) in the cohomology of the base. But since N_{2} is not orientable, it has a non-trivial first ℤ/2-characteristic class w_{1} = v_{1}. Since the first Wu class v_{1} is characterized by the relation Sq^{1}(x) = x · v_{1} for x a cohomology class of codimension 1, and since the codimension-1 cohomology is generated by v^{2j−1} and v^{2j}w, one of Sq^{1}(v^{2j−1}) and Sq^{1}(v^{2j}w) must be non-zero. But Sq^{1} is a derivation and Sq^{1}(v) = v^{2}, Sq^{1}(w) = w^{2} = 0, so

Sq^{1}(v^{2j}w) = (2j)v^{2j−1} Sq^{1}(v)w + v^{2j} Sq^{1}(w) = 0 + 0 = 0.

Thus Sq^{1}(v^{2j−1}) = (2j − 1)v^{2j−2} Sq^{1}(v) = v^{2j} ≠ 0. Since v^{2j} maps to 0 in the associated graded ring of H^{∗}(N, 𝔽_{2}), this forces v^{2j} = v^{2j−1}w (which shows incidentally that v_{1} = w, which one can check by other means). Thus v^{2j} pairs non-trivially with the ℤ/2-fundamental class [N_{2}]. Now a map f: N → Bπ = Bπ′ × Bℤ/2 is determined by choosing values for f^{∗} applied to a basis for H^{1}(Bπ, 𝔽_{2}) = H^{1}(Bπ′, 𝔽_{2})⊕H^{1}(Bℤ/2, 𝔽_{2}). So we define f by requiring that f^{∗}(x) = v, where 0 ≠ x ∈ H^{1}(Bℤ/2, 𝔽_{2}), and that f^{∗}: H^{1}(Bπ′, 𝔽_{2}) → H^{1}(N, 𝔽_{2}) lift g^{∗}: H^{1}(Bπ′, 𝔽_{2}) → H^{1}(M, 𝔽_{2}). From the calculation we just made, we see that for δ ∈ H^{∗}(Bπ′, 𝔽_{2}), we have

⟨δ · x^{k}, f_{∗}([N])⟩ = ⟨f^{∗}(δ · x^{k}), [N]⟩

= ⟨f^{∗}(δ) · v^{k}, [N]⟩

= \begin{cases}1, & k = 2j and ⟨δ, g_{∗}([N])⟩ ≠ 0, \\ 1, & k = 2j − 1 and ⟨f^{∗}(δ), [N_{1}]⟩ ≠ 0, \\ 0, & otherwise. \end{cases}

The fact that f_{∗}([N]) pairs non-trivially with δ · x^{2j}, where ⟨δ, g_{∗}([N])⟩ ≠ 0, shows that the class of N ⟶^{f} Bπ is not in the “tensor part” of the homology, and a dimension count shows that we get all the “Tor terms” by this construction. Now we just need to check that N has all the right properties. Since c preserves a metric of positive scalar curvature on M^{n−2j} × ℝℙ^{2j−1}, the product metric (of positive scalar curvature) on M^{n−2j} ×ℝℙ^{2j−1}×[0, 1] descends to a metric of positive scalar curvature on N (under the identification of M^{n−2j} × ℝℙ^{2j−1} × {0} with M^{n−2j} × ℝℙ^{2j−1} × {1}). The required orientation-reversing isometry is induced by c_{1} or c_{2}.

Now consider the exceptional case (b), i.e., the case of

H_{n−2}^{toral}(Bπ′, ℤ) ∗ H_{1}(Bℤ/2, ℤ).

In this case, the construction above does not yield a manifold of positive scalar curvature, so we need to do something different. Since H_{n−2}^{toral}(Bπ′, ℤ) is the image of the class of T^{n−2} → B(ℤ/2)^{n−2} under a map (ℤ/2)^{n−2} → π′ (cf. Remark 2.2(b)), it’s enough to suppose r = n − 1 and consider the homology Toda bracket

⟨z_{1}^{n−2}, 2, z_{1}⟩ = z_{1}^{n−3}⟨z_{1}, 2, z_{1}⟩.

(We have again used [2, 2.1, axiom 3].) If ⟨z_{1}, 2, z_{1}⟩ ∈ H_{3}(B(ℤ/2)^{2}, 𝔽_{2}) can be represented by a manifold P^{3} → ℝℙ^{∞}×ℝℙ^{∞} of positive scalar curvature, admitting an orientation-reversing isometry commuting up to homotopy with the map to ℝℙ^{∞} ×ℝℙ^{∞}, then T^{n−3} ×P^{3}, with the obvious map to Bπ, will satisfy the required conditions. But H_{3}(Bπ, ℤ) turns out to be spanned by z_{3} ⊗ 1 and 1 ⊗ z_{3} together with the class of the diagonal embedding Δ of ℝℙ^{3} in ℝℙ^{∞}, as one can see by observing that

Δ^{∗}: H^{∗}(Bπ, ℤ/2) = 𝔽_{2}[u, v] → H^{∗}(ℝℙ^{3}, ℤ/2) = 𝔽_{2}[x]/(x^{4})

sends both of the generators u and v to x, so that Δ_{∗}([ℝℙ^{3}]) pairs non-trivially with both u^{2}v and v^{2}u, and thus can’t be in the span of the classes z_{3} ⊗ 1 and 1 ⊗ z_{3}. Thus Δ_{∗}(z_{3}) can be chosen as a representative for a generator of Tor_{ℤ}(H_{1}(Bℤ/2, 𝔽_{2}), H_{1}(Bℤ/2, 𝔽_{2})), and it clearly satisfies our requirements. This completes the inductive step. □

Proof of Theorem 2.4. Theorem 2.4 now follows immediately from combining Theorem 5.1 and Theorem 3.1(2). □

Remark 5.2 (added after this paper was accepted). Michael Joachim has now shown [6] that if π is an elementary abelian 2-group, every toral class in H_{n}(Bπ, ℤ), n ≥ 3, is represented by an oriented manifold of positive scalar curvature. Combining this result with Theorem 2.4 yields the following stronger result:

Theorem 5.3. Let π be an elementary abelian 2-group, and let n ≥ 5. Then every oriented n-manifold with fundamental group π and with non-spin universal cover has a metric of positive scalar curvature.

In other words, the Gromov-Lawson-Rosenberg Conjecture holds for oriented manifolds with non-spin universal cover and fundamental group an elementary abelian 2-group.

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Department of Mathematics, University of Oregon, Eugene, OR 97403-1222

E-mail address: botvinn@poincare.uoregon.edu
URL: http://darkwing.uoregon.edu/~botvinn

Department of Mathematics, University of Maryland, College Park, MD 20742-4015

E-mail address: jmr@math.umd.edu
URL: http://www.math.umd.edu/~jmr