On Invariants of Hirzebruch and Cheeger-Gromov

Stanley Chang^{1} and Shmuel Weinberger^{2}

In their analysis of free involutions on the sphere, Browder and Livesay [BL] introduced an invariant which they used to study the existence and uniqueness question for desuspensions. In particular, they produced infinitely many manifolds homotopy equivalent, but not diffeomorphic or even homeomorphic, to ℝℙ^{4k−1} for k > 1. Hirzebruch [Hi] soon gave an alternative definition of this invariant as follows: Any manifold M of dimension 4k − 1 with π_{1}(M) = ℤ_{2} has a multiple ℓM which bounds some W^{4k} such that π_{1}(M) → π_{1}(W) is an isomorphism. Define

τ(M) = \frac{1}{ℓ} · g − sig (W̃) = \frac{1}{ℓ} ( tr g |_{H_{+}} − tr g |_{H_{−}}) = \frac{1}{ℓ} ( sig (W̃) − 2 sig (W)) ,

where g ∈ ℤ_{2} is the nontrivial element and W̃ is the two-fold cover of W with H_{±} ≡ H_{±}^{2k}(W̃). The fact that τ(M) is independent of W can be seen either using the G-signature formula [APS] or by a bordism argument.

Remark: Actually, Hirzebruch’s invariant was only shown a bit later to be equal (up to a sign) to the Browder-Livesay invariant by Hirzebruch and Lopez de Medrano [LdM].

It is quite simple to modify τ to obtain invariants of arbitrary M^{4k−1} with finite fundamental group G = π_{1}(M) by using the G-signature of W̃ modulo the regular representation (as in [Wa]) or by using characters equal to tr g |_{H_{+}} − tr g |_{H_{−}} for any g ≠ e. As a very special and important case, one can even simply use sig (W̃)−|G| sig (W) which equals ∑_{g≠e} χ_{g}(sig_{G}(W)) to obtain a quite useful invariant.

As an application of their index theorem for manifolds with boundary, Atiyah, Patodi and Singer [APS] obtain such invariants of M without introducing W. More precisely, the APS invariant requires the use of an auxiliary Riemannian metric on M, but their index theorem for manifolds with boundary shows that the number obtained is independent of this metric. All of the numerical invariants mentioned above are differences of η-invariants (which measure “spectral asymmetry”) for the signature operator twisted by suitable flat bundles.

The APS method admits extensive generalization to the case in which π_{1}(M) is possibly infinite. In [A] Atiyah generalized the index theorem from compact closed manifolds to universal covers of compact manifolds (or even manifolds with proper actions of a discrete group). Using von Neumann traces, he defined

ind_{Γ}( ˜/ D) = dim_{Γ}(ker ˜/ D on L^{2}(M̃)) − dim_{Γ}(coker ˜/ D on L^{2}(M̃)).

Both [L1] and [L2] are excellent references for ideas related to the von Neumann trace and its applications in geometry and topology.

1Research partially supported by NSF Grant DMS-9971657.

2Partially supported by an NSF grant.

Note: Atiyah showed that ind_{Γ}( ˜/ D) = ind ( / D), which generalizes the multiplicativity property of ordinary indices for finite coverings (see appendix). This result is quite remarkable since, if the group is infinite, the elements of ker / D never lift to be L^{2} on the universal cover, but they nonetheless lead indirectly to elements of the L^{2} kernel (at the level of indices, of course). Cheeger and Gromov [CG1,2] introduced the analogue of the APS invariant in this setting: namely they studied an L^{2}-invariant η_{(2)}(M̃) of compact M and the difference ρ_{(2)}(M) = η_{(2)}(M̃) − η(M) which is metric independent. For Γ = ℤ_{2} the quantity ρ_{(2)}(M) is simply the Hirzebruch invariant or the Browder-Livesay invariant.

Thom’s classical work on cobordism implies that every compact odd dimensional oriented manifold M has a multiple rM which is the boundary of an oriented manifold W. Hausmann [Ha] showed furthermore that, for every such M with fundamental group Γ, there is a manifold W such that Γ injects into π_{1}(W) and ∂W = rM, for some multiple rM of M; that is, he showed that nullcobordant manifolds bound in such a way that their fundamental groups inject. If Γ is a group, we can define the L^{2}-signature sig _{(2)}^{Γ}(V) = dim_{Γ}(V^{+}) − dim_{Γ}(V^{−}) for any ℓ^{2}(Γ)-module V endowed with nonsingular symmetric bilinear form. As usual, one uses an inner product to rewrite the symmetric bilinear form in terms of a self-adjoint operator, and then uses the spectral theorem to obtain projections to the positive and negative definite part. If N^{4k} is a Γ-space, let sig _{(2)}^{Γ}(N) be given by the L^{2}-signature of the symmetric form induced by cap product on its middle cohomology.

Define a new “Hirzebruch type” invariant τ_{(2)} given by

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

where G = π_{1}(W) and W_{G} is the induced G-cover of W. Suppose that rM = ∂W with π_{1}(M) = Γ injecting into π_{1}(W) = G. If G injects into some larger group G′, and let W_{G′} be the G′-space induced from the G action on W_{G} to G′. Notice that ∂(W_{G′}/G′) = rM, so we can use the larger group G′ to define τ_{(2)}. However, by the Γ-induction property of Cheeger-Gromov [CG2, 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) − sig (W))
= τ_{(2)}^{G}(M).

So one can pass to any larger group without changing the value of the invariant. Now, given two manifolds W and W′ with the required bounding properties, we can use the large group G′ = π_{1}(W) ∗_{Γ} π_{1}(W′), which contains both fundamental groups, and the usual Novikov additivity argument, to see that τ_{(2)} is independent of all choices.

In fact, according to [LS1], we have τ_{(2)} = ρ_{(2)}, but we do not need this equality for the proofs of our main theorems. The goal of this paper is to prove a few theorems about τ_{(2)}(M) (or ρ_{(2)}(M)) that extend or are analogues of the classically understood situation for finite fundamental group.

Theorem 1: For any compact oriented manifold M^{4k−1}, where k ≥ 2, such that π_{1}(M) is not torsion-free, there exist infinitely many manifolds M_{i} which are simple homotopy equivalent and tangentially equivalent to M, but not homeomorphic to M. If M is smooth, the M_{i} can be taken to be smooth as well. This infinite number is detected by τ_{(2)}.

Remarks:

1. Very similar arguments to the proof of Theorem 1 show that any M^{4k+2} with non-torsion-free fundamental group has infinite diffeotopy group.

2. The analogue for M^{4k+1} is false. For instance, the real projective space ℝℙ^{4k+1} has a finite structure set. See [LdM]. Also, we cannot dispense with the orientability hypothesis as ℝℙ^{2} × S^{5} has finite structure set.

3. There are predictions in the literature about L_{4k}(Γ) ⊗ ℚ and its expression in terms of the group homology of normalizers of finite subgroups. See [FJ] and page 261 of [We2]. These would suggest that Theorem 1 is true except for the thorny problem of self-homotopy equivalences of M. These maps h : M → M could represent nontrivial elements of the structure set S(M), but nonetheless they are not the elements described in Theorem 1.

4. If M is smooth, all the M_{i} can be taken smooth as well, without even being homeomorphic. The extension of τ_{(2)} from smooth to topological manifolds is not very hard by bordism methods. After multiplication by a positive integer, every topological manifold is cobordant to a smooth manifold N via a cobordism V (all with the same fundamental group), so we can consider τ_{(2)}(N) − (sig_{(2)} − sig )(V).

5. The theorem above asserts that, whenever π_{1}(M) has torsion, then τ_{(2)} is not a homotopy invariant. Mathai [Ma] conjectured that the converse is true for ρ_{(2)} and proved it for torsion-free crystallographic fundamental group. Thus, the above result is the converse to Mathai’s conjecture.

Mathai’s conjecture is an analogue of a conjecture and theorem of the second author [We1] for Atiyah-Patodi-Singer invariants. Keswani [K] proved a general homotopy invariance theorem: if the assembly map K_{∗}(BΓ) → K_{∗}(C∗_{max}Γ) is surjective, then ρ_{(2)} is a homotopy invariant for manifolds with fundamental group Γ. This hypothesis holds for fundamental group of real and complex hyperbolic manifolds for torsion-free amenable groups by the theorem of Higson and Kasparov [HK]. However, it fails for all nontrivial groups satisfying property T, i.e. for many torsion-free lattices in higher rank or in Sp (n, 1). Our next theorem somewhat repairs this problem.

Theorem 2: If Γ is a torsion-free discrete subgroup of SL_{n}(ℝ), then ρ is a homotopy invariant for manifolds with fundamental group Γ. More generally, if Γ = π_{1}(M) is residually finite and the Borel conjecture holds for Γ, then ρ_{(2)} is a homotopy invariant (see e.g. [FJ]).

In a future paper, the first author plans to remove the residual finiteness hypotheses arising in this work.

We now turn to the proof of Theorem 1. Let M be as in the theorem and ℤ_{n} ⊂ π_{1}(M) = Γ be a nontrivial cyclic subgroup. The M_{i} are just the results of acting by elements of im (L_{4k}(ℤ_{n}) → L_{4k}(Γ)) on S(M) in the surgery exact sequence

⋯ → H_{4k}(M, 𝕃) → L_{4k}(Γ) → S(M) → H_{4k−1}(M, 𝕃) → L_{4k−1}(Γ) → ⋯

The image of A : H_{4k}(M, 𝕃) → L_{4k}(Γ) consists of quadratic forms arising from closed manifolds and for such manifolds we have sig _{(2)}^{Γ}(N) = sig (N) by Atiyah [A]. Note that this

higher signature may be viewed as a map sig _{(2)}^{Γ} : L_{4k}(Γ) → ℝ (see [Wa] for the definition

of L-groups). This image must equal ker (L_{4k}(Γ) → S(M)), the set of all a ∈ L_{4k}(Γ) for which a(M) = M diffeomorphically.^{3} If a ∈ L_{4k}(Γ), let V_{a} be the form representing it.

Let α_{Γ} : L_{4k}(Γ) → ℝ be given by the homomorphism α_{Γ}(V) ≡ sig _{(2)}^{Γ}(V) − sig (V). Hence, if a(M) = M, then α_{Γ}(V_{a}) = 0. We would like to produce infinitely many V_{a′} such that α_{Γ}(V_{a′}) ≠ 0.

To show that the homomorphism α_{Γ} : L_{4k}(Γ) → ℝ has infinite image, it suffices to show that it is nontrivial. Let p_{∗} : L_{4k}(ℤ_{n}) → L_{4k}(Γ) be the map induced by the injection p : ℤ_{n} → Γ. By the induction property of Γ-dimension, we have the commutative diagram

\xymatrix{L_{4k}(ℤ_{n}) \ar[d]_{p_{∗}} \ar[r]^{^{^{sig}^{(2)}}^{ℤn}} & ℝ \\ L_{4k}(Γ) \ar[ur]_{_{_{sig}_{(2)}}_{Γ}}}

so we only have to check that α_{ℤ_{n}} : L_{4k}(ℤ_{n}) → ℝ is nontrivial. We note that, inverting 2, it makes no difference whether the coefficient ring is ℤ or ℚ [Ran] or whether one works with projective, free or based-free modules (by the well-known “Rothenberg exact sequences”). Thus, we need only produce a bilinear form on a projective module over ℚℤ_{n}, namely the one-by-one bilinear form [1] on the module ℚ, which is projective over ℚℤ_{n}. By definition, our invariant is \frac{1}{n} − 1 for this element, which is always nontrivial for n > 1.

If rM cobounds some space W for which π_{1}(M) injects into G = π_{1}(W), we can extend each boundary component M by the cobordism Y, so that rM′ cobounds W′, where W′ is obtained from W and r copies of Y. Hence by Novikov additivity and Γ-induction, we have

τ_{(2)}^{G}(M) − τ_{(2)}^{G}(M′) = \frac{1}{r}(sig _{(2)}^{G}(W_{G}) − sig (W)) − \frac{1}{r}(sig _{(2)}^{G}(W′_{G}) − sig (W′))
= \frac{1}{r}(sig _{(2)}^{G}(W_{G}) − sig (W)) − \frac{1}{r}(sig _{(2)}^{G}(W) + r · sig _{(2)}^{G}(Y_{G}) − sig (W) − r · sig (Y))
= sig _{(2)}^{G}(Y_{G}) − sig (Y)
= sig _{(2)}^{Γ}(Y_{Γ}) − sig (Y)
= α_{Γ}(V),

3More precisely, the equality represents a diffeomorphism homotopic to the homotopy equivalence implicit in the fact that a(M) is a member of a structure set.

where V ∈ L_{4k}(Γ) represents the cobordism Y.

Let V_{a} ∈ L_{4k}(Γ) such that α_{Γ}(V_{a}) ≠ 0, where Γ = π_{1}(M). Since (M → M) ∈ S_{4k−1}(M), there is a cobordism Y_{a} such that ∂Y_{a} = M ∪ M′ representing V_{a}. Clearly M and M′ are homotopy equivalent but not diffeomorphic. Consider V_{a_{i}} ∈ L_{4k}(Γ) for all i ∈ ℕ such that α_{Γ}(V_{a_{i}}) ≠ α_{Γ}(V_{a_{j}}) for all i ≠ j and all nonzero. Let M_{i} be the result of acting on M by V_{a_{i}}. Then τ_{(2)}^{G}(M) − τ_{(2)}^{G}(M_{i}) = α_{Γ}(V_{a_{i}}) ≠ 0. Also

\begin{aligned}τ_{(2)}^{G}(M_{i}) − τ_{(2)}^{G}(M_{j}) & = & τ_{(2)}^{G}(M_{i}) − τ_{(2)}^{G}(M) + τ_{(2)}^{G}(M) − τ_{(2)}^{G}(M_{j}) \\ & = & −α_{Γ}(V_{α_{i}}) + α_{Γ}(V_{α_{j}}) ≠ 0, \end{aligned}

as desired. •

Now we prove Theorem 2. For a finite group Γ, the rho-invariant ρ_{(2)} = η_{(2)} − η is equal to \frac{1}{|Γ|} η_{⊗ℝΓ} − η, where η_{⊗ℝΓ} is the eta-invariant for M with coefficients in the flat bundle associated to the regular representation. According to [LS2], if Γ is residually finite with descending quotients Γ_{k}, then

ρ_{(2)} = lim_{k→∞} (\frac{1}{|Γ_{k}|} (η_{⊗ℝΓ} − η)) .

Indeed, Cheeger and Gromov introduced ρ_{(2)} to generalize the right-hand side to non-residually finite fundamental group. However, according to [We2], for any representation V of a group satisfying the Borel conjecture, the quantity \frac{1}{dim V} η_{⊗V} − η is a homotopy invariant. Since ρ is the limit of homotopy invariants, a fortiori, it too is a homotopy invariant. •

If Γ is a discrete subgroup of SL_{n}(ℝ), then it is a classical lemma of Selberg that Γ is residually finite. The fact that the Borel conjecture holds if Γ is in addition torsion-free is the main result of [FJ].

Problem: If π_{1}(M) has many conjugacy classes of elements of finite order, then one would expect that S(M) is rather larger than the ℤ we just detected. Can this statement be proven unconditionally? Wolfgang Lück has pointed out that, using the traces associated to elements of finite order that have finitely many conjugates (e.g. central elements), one can extend Theorem 1 and detect a larger S(M). But this result is still rather smaller than the predicted size of the structure set.

Appendix: Acyclic groups and Atiyah’s theorem

For the convenience of the reader we shall prove here the theorems of Atiyah and Hausmann invoked in the paper. Let M be a closed manifold. According to [BDM], there is an acyclic group A with an injective homomorphism h: π_{1}(M) → A. Since A is acyclic, the map Ω_{n}(K(A, 1)) → Ω_{n}(∗) is an isomorphism, so that, if M bounds (i.e. M is zero in Ω_{n}(∗)), it bounds a manifold W such that

\xymatrix{M \ar[d] \ar[r] & K(π_{1}(M), 1) \ar[d] \\ W \ar[r] & K(A, 1)}

commutes; a fortiori π_{1}(M) → π_{1}(W) is injective. This argument gives Hausmann’s result.

Now for Atiyah’s theorem we observe, by the induction and bordism invariance properties of sig_{(2)}, that we have the commutative diagram

\xymatrix{ & Ω_{4k}(∗) \ar[dl]_{sig} \ar[d] \ar[dr]^{_{_{sig}_{(2)}}} \\ ℤ & Ω_{4k}(K(π, 1)) \ar[l]^{sig} \ar[d] \ar[r]_{_{_{sig}_{(2)}}_{π}} & ℝ \\  & Ω_{4k}(K(A, 1)) \ar[ul]^{sig} \ar[ur]_{_{_{sig}_{(2)}}_{A}}}

Since Ω_{4k}(∗) → Ω_{4k}(K(A, 1)) is an isomorphism and sig = sig_{(2)} for the trivial group, the conclusion follows.

We would like to thank both Wolfgang Lück and the referee for their helpful comments.

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Department of Mathematics, Wellesley College, Wellesley, MA 02481.

Department of Mathematics, University of Chicago, Chicago, IL 60637.