On conjectures of Mathai and Borel

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 = π1(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 = π1(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 μmax : K0(BΓ) → K0(CΓ) is an isomorphism. Yet it is now known that μmax 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 manifoldM with fundamental group Γ, there is a manifoldW with boundary such that Γ injects into G = π1(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 M4k−1 given by τ(2)(M) = 1 r ( sig(2)(W̃ )− sig (W )), where W̃ is the universal cover of W . The map sig(2) is a real-valued homomorphism on the L-theory group L4k(G) given by sig(2)(V ) = dimG(V +)− dimG(V −) for any quadratic form V , considered as an `2(G)-module. This invariant τ(2) is in general a diffeomorphism invariant, but is not a homotopy invariant when π1(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 M4k−1 if the fundamental group Γ = π1(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. Research partially supported by NSF Grant DMS-9971657.