On conjectures of Mathai and Borel

Stanley Chang^{1}

Abstract. Mathai [M] has conjectured that the Cheeger-Gromov invariant ρ_{(2)} = η_{(2)} − η is a homotopy invariant of closed manifolds with torsion-free fundamental group. In this paper we prove this statement for closed manifolds M when the rational Borel conjecture is known for Γ = π_{1}(M), i.e. the assembly map α: H_{∗}(BΓ, ℚ) → L_{∗}(Γ)⊗ℚ is an isomorphism. Our discussion evokes the theory of intersection homology and results related to the higher signature problem.

Let M be a closed, oriented Riemannian manifold of dimension 4k − 1, with k ≥ 2. In [Ma] Mathai proves that the Cheeger-Gromov invariant ρ_{(2)} ≡ η_{(2)} − η is a homotopy invariant of M if Γ = π_{1}(M) is a Bieberbach group. In the same work, he conjectures that ρ_{(2)} will be a homotopy invariant for all such manifolds M whose fundamental group Γ is torsion-free and discrete. This conjecture is verified by Keswani [K] when Γ is torsion-free and the Baum-Connes assembly map μ_{max} : K_{0}(BΓ) → K_{0}(C^{∗}Γ) is an isomorphism. Yet it is now known that μ_{max} fails to be an isomorphism for groups satisfying Kazhdan’s property T. This paper improves on Keswani’s result by showing that Mathai’s conjecture holds for torsion-free groups satisfying the rational Borel conjecture, for which no counterexamples have been found.

As a consequence of a theorem of Hausmann [H], for every compact odd-dimensional oriented manifold M with fundamental group Γ, there is a manifold W with boundary such that Γ injects into G = π_{1}(W) and ∂W = rM for some multiple rM of M. Using this result, the author and Weinberger construct in [CW] a well-defined Hirzebruch-type invariant for M^{4k−1} given by

τ_{(2)}(M) = \frac{1}{r} ( sig _{(2)}^{G}(W̃) − sig (W)),

where W̃ is the universal cover of W. The map sig _{(2)}^{G} is a real-valued homomorphism on the L-theory group L_{4k}(G) given by sig _{(2)}^{G}(V) = dim_{G}(V^{+}) − dim_{G}(V^{−}) for any quadratic form V, considered as an ℓ^{2}(G)-module. This invariant τ_{(2)} is in general a diffeomorphism invariant, but is not a homotopy invariant when π_{1}(M) is not torsion-free [CW]. It is now also known that τ_{(2)} coincides with ρ_{(2)} by the work of Lück and Schick [LS].

The purpose of this paper is to show that the diffeomorphism invariant τ_{(2)}, and subsequently ρ_{(2)}, is actually a homotopy invariant of M^{4k−1} if the fundamental group Γ = π_{1}(M) satisfies the rational Borel conjecture, which states that the assembly map α: H_{∗}(BΓ, ℚ) → L_{∗}(Γ)⊗ℚ is an isomorphism if Γ is torsion-free. We include in our discussion some background in L-theory and intersection homology.

I would like to thank Shmuel Weinberger for some useful conversations.

1Research partially supported by NSF Grant DMS-9971657.

Introduction

The classical work on cobordism by Thom implies that every compact odd-dimensional oriented manifold M has a multiple rM which is the boundary of an oriented manifold W. Hausmann [H] showed furthermore that, for every such M with fundamental group Γ, there is a manifold W such that ∂W = rM, for some multiple rM of M, with the additional property that the inclusion M ↪ W induces an injection Γ ↪ π_{1}(W). For such a manifold M^{4k−1} with fundamental group Γ, a higher “Hirzebruch type” real-valued function τ_{(2)} is defined in [CW] in the following manner:

τ_{(2)}^{G}(M) = \frac{1}{r} ( sig _{(2)}^{G}(W̃) − sig (W)),

where G = π_{1}(W) and W̃ is the universal cover of W.

To see that this quantity can be made independent of (W, G), we consider any injection G ↪ G′. Let W_{G′} be the G′-space induced from the G-action on W̃ to G′. Now define

τ_{(2)}^{G′}(M) = \frac{1}{r} ( sig _{(2)}^{G′}(W_{G′}) − sig (W_{G′}/G′)).

Since W̃ is simply the G-cover W_{G} of W and the quotient W_{G′}/G′ is clearly diffeomorphic to W, we note that this definition of τ_{(2)}^{G′} is consistent with the above definition of τ_{(2)}^{G}. However, by the Γ-induction property of Cheeger-Gromov [CG, page 8, equation (2.3)], we have

τ_{(2)}^{G′}(M) = \frac{1}{r} ( sig _{(2)}^{G′}(W_{G′}) − sig (W_{G′}/G′))
= \frac{1}{r} ( sig _{(2)}^{G}(W_{G}) − sig (W))
= τ_{(2)}^{G}(M).

So from G one can pass to any larger group G′ without changing the value of this quantity. Given two manifolds W and W′ with the required bounding properties, we can use the larger group G′ = π_{1}(W) ∗_{Γ} π_{1}(W′), which contains both fundamental groups π_{1}(W) and π_{1}(W′). The usual Novikov additivity argument^{2} proves that τ_{(2)}^{G}(M) is independent of all choices [CW]. We will henceforth refer to it as τ_{(2)}.

In [CW] we prove that τ_{(2)} is a differential invariant but not a homotopy invariant of M^{4k−1} when π_{1}(M) is not torsion-free. We will use ideas given by Weinberger in [W] which under the same conditions proves the homotopy invariance of the twisted ρ-invariant ρ_{α}(M) = η_{α}(M) − η(M), where α is any representation π_{1}(M) → U(n). We supply brief expository remarks about intersection homology and algebraic Poincaré complexes in Sections I and II before proving the theorem in Section III.

2Novikov additivity, proved cohomologically in [AS], posits that signature is additive in the following sense: if Y is an oriented manifold of dimension 2n with boundary X and Y′ is another such manifold with boundary −X, then sig (Y ∪_{X} Y′) = sig (Y) + sig (Y′). The additivity for sig _{(2)}^{G} is easy to argue on the level of ℓ^{2}(G)-modules V endowed with nonsingular bilinear form. To see that this additivity corresponds to appropriate manifold glueings, we refer the reader to [F1] and [F2] of Farber, who puts L^{2} cohomology groups into a suitable framework in which the same arguments can be repeated.

I. Intersection homology

Following Goresky and MacPherson, we say that a compact space X is a pseudomanifold of dimension n if there is a compact subspace Σ with dim(Σ) ≤ n − 2 such that X − Σ is an n-dimensional oriented manifold which is dense in X. We assume that our pseudomanifolds come equipped with a fixed stratification by closed subspaces

X = X_{n} ⊃ X_{n−1} = X_{n−2} = Σ ⊃ X_{n−3} ⊃ ⋯ ⊃ X_{1} ⊃ X_{0}

satisfying various neighborhood conditions (see [GM]). A perversity is a sequence of integers \overline{p} = (p_{2}, p_{3}, ⋯, p_{n}) such that p_{2} = 0 and p_{k+1} = p_{k} or p_{k} + 1. A subspace Y ⊂ X is said to be (\overline{p}, i)-allowable if dim(Y) ≤ i and dim(Y ∩ X_{n−k}) ≤ i − k + p_{k}. We denote by IC\overline{p} ; i the subgroup of i-chains ξ ∈ C_{i}(X) for which |ξ| is (\overline{p}, i)-allowable and |∂ξ| is (\overline{p}, i − 1)-allowable. If X is a pseudomanifold of dimension n, then we define the i-th intersection homology group IH\overline{p} ; i(X) to be the i-th homology group of the chain complex IC\overline{p} ; ∗(X). For any perversity \overline{p}, we have

IH\overline{p} ; 0(X) ≅ H^{n}(X) and IH\overline{p} ; n(X) ≅ H_{n}(X).

Whenever \overline{p} + \overline{q} ≤ \overline{r}, the intersection homology groups can be equipped with a unique product ∩: IH\overline{p} ; i(X) × IH\overline{q} ; j(X) → IH\overline{r} ; i+j−n(X) that respects the intersection homology classes of dimensionally transverse pairs (C, D) of cycles. Let \overline{m} = (0, 0, 1, 1, 2, 2, …, 2k−2, 2k−2, 2k−1) be the middle perversity. If X is stratified with only strata of even codimension, and if dim(X) = 4k, then the intersection pairing

∩: IH\overline{m} ; 2k(X) × IH\overline{m} ; 2k(X) → ℤ

is a symmetric and nonsingular form when tensored with ℚ. We can define the signature sig (X) of X to be the signature of this quadratic form. If X is a manifold, this definition coincides with the usual notion of signature.

If X is a pseudomanifold, we say that X is a Witt space if IH\overline{m} ; k(L, ℚ) = 0 whenever L^{2k} is the link of an odd-codimensional stratum of X. If X^{q} is a Witt space, there is a nondegenerate rational pairing

IH\overline{m} ; i(X, ℚ) × IH\overline{m} ; j(X, ℚ) → ℚ

whenever i + j = q. If q = 4k > 0, then IH\overline{m} ; 2k(X, ℚ) is a symmetric inner product space. In this case, we define the Witt class w(X) of X to be the equivalence class of IH\overline{m} ; 2k(X, ℚ) in the Witt ring W (ℚ) of classes of symmetric rational inner product spaces. If q = 0, set w(X) to be rank (H_{0}(X, ℚ)) · ⟨1⟩, and set it to zero if q ≢ 0 mod 4. If (X, ∂X) is a Witt space with boundary, set w(X) = w(X̂), where X̂ = X ∪ cone(∂X). Let sig_{ℚ}: W (ℚ) → ℤ be the signature homomorphism developed by Milnor and Husemoller [MH]. We define the signature sig (X) to the integer sig_{ℚ}(w(X)). See [S].

Denote by Ω∗^{Witt} the bordism theory based on Witt spaces; i.e. if Y is a Witt space, define Ωn^{Witt}(Y) to be the classes [X, f], where X is an n-dimensional Witt space and f: X → Y a continuous map, such that [X_{1}, f_{1}] ∼ [X_{2}, f_{2}] iff there is an (n + 1)-dimensional Witt space W with ∂W = X_{1} ∐ X_{2} and a map W → Y that restricts to f_{1} and f_{2} on the boundary. Witt bordism enjoys the important properties that (1) there is a signature invariant defined on the cycle level which is a cobordism invariant, and (2) the signature can be extended to relative cycles (X, ∂X) so that it is additive.

Lemma 1: Let M_{1} and M_{2} be homotopy equivalent manifolds of the same dimension. Suppose that their fundamental group Γ is torsion-free and satisfies the rational Borel conjecture. Then there is rationally a Witt cobordism between M and M′ over BΓ.

Proof: It is well-known that, if Γ is torsion-free and satisfies the Borel conjecture, then it satisfies the Novikov conjecture, which asserts that, if f: M → BΓ is a map, then the generalized Pontrjagin number (or “higher signature”)

f_{∗} (L(M) ∩ [M]) ∈ H_{∗}(BΓ, ℚ)

is an oriented homotopy invariant. Here L(M) is the Hirzebruch L-polynomial in terms of the Pontrjagin classes, and [M] is the fundamental class of M. Let f_{i}: M_{i} → BΓ be the map classifying the universal cover of M_{i}. We would like to show that [M_{1}, f_{1}] and [M_{2}, f_{2}] rationally define the same class in Ω∗^{Witt}(BΓ).

For a Witt space X, Siegel [S] constructs an L-class L(X) ∈ H_{∗}(X, ℚ) that generalizes the L-class of Goresky and MacPherson [GM], the latter of which is defined only on Whitney stratified pseudomanifolds with even-codimensional strata. If f: X → BΓ is the universal cover of X, then f_{∗}(L(X)) ∈ H_{∗}(BΓ, ℚ) coincides with the higher signature given above. In addition, it is a bordism invariant in Ω∗^{Witt}(BΓ). Rationally the Atiyah-Hirzebruch spectral sequence for Witt cobordism collapses, so we obtain an isomorphism

i: Ωn^{Witt}(BΓ) ⊗ ℚ ⟶ ⨁_{k≥0} H_{n−4k}(BΓ, ℚ)

such that i([M, f]) is the higher signature f_{∗}(L(M) ∩ [M]). But these signatures are homotopy invariants by assumption. ■

II. Algebraic Poincaré complexes

An n-dimensional Poincaré complex over a ring A with involution is an A-module chain complex C with a collection of A-module morphisms ϕ_{s}: C^{n−r+s} → C_{r} such that the chain map ϕ_{0} is a chain equivalence inducing abstract Poincaré duality isomorphisms ϕ_{0} : H^{n−r}(C) → H_{r}(C). As defined by Wall [Wa], an n-dimensional geometric Poincaré complex X with fundamental group Γ is a finitely dominated CW-complex together with a fundamental class [X] ∈ H_{n}(X̃, ℤ) such that cap product with [X] gives a family of ℤΓ-module isomorphisms

∩ [X]: H^{r}(X̃) ⟶ H_{n−r}(X̃)

for 0 ≤ r ≤ n (there is additional business about an orientation group morphism w: Γ_{1}(X) → ℤ_{2} for which one should consult [Ran2]). An n-dimensional geometric Poincaré complex X with fundamental group Γ naturally determines an n-dimensional symmetric Poincaré complex (C(X̃), ϕX̃) over ℤΓ. Such symmetric complexes (as opposed to quadratic complexes, which define lower L-theory) over a ring A can be assembled into an abelian group L^{n}(A) under a cobordism relation defined by abstract Poincaré-Lefschetz duality [M,Ran1], with addition given by

(C, ϕ) + (C′, ϕ′) = (C ⊕ C′, ϕ ⊕ ϕ′) ∈ L^{n}(A).

For a more detailed discussion on algebraic Poincaré complexes and in particular the manner in which algebraic Poincaré complexes are glued together along a common boundary, see [Ran3].

Lemma 2: Let M and M′ be homotopy equivalent manifolds of the same dimension 4k −1 with torsion-free fundamental group Γ. Suppose that Y is a rational Witt cobordism of M and M′ over BΓ. If the Borel conjecture holds for Γ, then sig _{(2)}^{Γ}(Y_{Γ}) = sig (Y), where Y_{Γ} is the induced Γ-cover of Y.

Proof: By the preceding lemma, there is a Witt cobordism between n copies of M and n copies of M′. Let C^{h} be the homotopy cylinder given by the homotopy equivalence h : M → M′. Attach n copies of C^{h} to the rational Witt cobordism Y to form a space X. This space is usually not a pseudomanifold because the singular space is of codimension one, but it is an algebraic Poincaré space. Notice that all of these spaces come equipped with a map to BΓ. Since upper and lower L-theory coincide rationally, we may then consider X as an element of L_{∗}(BΓ) ⊗ ℚ. Consider then the composition of maps

Ω∗^{SO}(BΓ) ⊗ ℚ ⟶ H_{∗}(BΓ, ℚ) ⟶ L_{∗}(Γ) ⊗ ℚ .

The first map is well known to be surjective, and the second is surjective by our assumption that the Borel conjecture holds for Γ. Therefore there is a closed manifold X′ without boundary bordant to X as algebraic Poincaré complexes. Atiyah’s Γ-index theorem [A] shows that sig _{(2)}^{Γ}(X′_{Γ}) = sig (X′), so that the cobordism invariance of signature implies that sig _{(2)}^{Γ}(X_{Γ}) = sig (X). For a topological proof of the Γ-index theorem for signature, see the appendix of [CW].

By Novikov additivity, we may decompose the complex X to arrive at the equation

sig _{(2)}^{Γ}(Y_{Γ}) + n · sig _{(2)}^{Γ}(C_{Γ}^{h}) = sig (Y) + n · sig (C^{h}).

By Novikov additivity, the homotopy cylinder C^{h} enjoys the convenient property that sig _{(2)}^{Γ}(C_{Γ}^{h}) = sig (C^{h}), so sig _{(2)}^{Γ}(Y_{Γ}) = sig (Y). ■

III. The main theorem

Theorem: Let M be an oriented compact manifold of dimension 4k−1. If π_{1}(M) is torsion-free, then the real-valued quantity τ_{(2)}(M) is a homotopy invariant.

Proof: Let M and M′ be homotopy equivalent closed manifolds of dimension n = 4k − 1. There are closed manifolds W and W′ such that ∂W = rM and ∂W′ = sM′ with injecting fundamental groups. Construct s copies of W and r copies of W′ and attach rs Witt cobordisms Y between the boundary components M and M′. Call this space X with H = π_{1}(X). Note that both π_{1}(W) and π_{1}(W′) inject into H. Since the map Ω^{SO}(BH) ⊗ ℚ → Ω^{Witt}(BH) ⊗ ℚ is surjective (see [Cu]), it follows that X is Witt cobordant to a smooth manifold X′. Hence they share the same signatures. Note that X′ has the property that sig _{(2)}^{H}(X′_{H}) = sig (X′), and hence sig _{(2)}^{Γ}(X′_{Γ}) = sig (X′) by Atiyah’s Γ-index theorem. By the observation above, we can then conclude that sig _{(2)}^{Γ}(X_{Γ}) = sig (X).

By Novikov additivity, we can partition X along the original attachments to obtain s · sig _{(2)}^{Γ}(W_{Γ}) + rs · sig _{(2)}^{Γ}(Y_{Γ}) − r · sig _{(2)}^{Γ}(W_{Γ}) = s · sig (W) + rs · sig (Y) − r · sig (W′).

By Lemma 2, the cylindrical terms cancel, leaving

\frac{1}{r} (sig _{(2)}^{Γ}(W_{Γ}) − sig (W)) = \frac{1}{s} (sig _{(2)}^{Γ}(W′_{Γ}) − sig (W′)) .

Equivalently τ_{(2)}(M) = τ_{(2)}(M′), so τ_{(2)} is a homotopy equivalence. ■

Remark: Given the identification of τ_{(2)} and ρ_{(2)}, one can conjecture that ρ_{(2)} is homotopy invariant iff the fundamental group of M is torsion-free (see [CW]). However, recent work of [NW] makes use of secondary invariants for groups with unsolvable word problems, which, a fortiori, are not residually finite. Potentially, the results of this paper can have applications to the geometry of certain moduli spaces.

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