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This special case of the generalized Baum-Connes Conjecture is usually known as the Connes-Kasparov Conjecture, and has been proved for connected linear reductive Lie groups by Wassermann [Was]. , [Pl3], [BHP], [BCH]). For arbitrary closed subgroups G of amenable connected Lie groups, of SO(n, 1), or of SU (n, 1), the generalized Baum-Connes Conjecture follows from still stronger results of Kasparov et al. ([Kas4], [Kas5], [Kas7], [JuK]). ) Around 1980, Gromov and Lawson ([GL1], [GL2]) began to notice an interesting parallel between the Novikov Conjecture and a problem in Riemannian geometry, that of determining what smooth manifolds admit Riemannian metrics of positive scalar curvature.

Without great loss of generality, suppose the dimension n of M is even, n = 2k. We compare a certain analytic invariant of M , which one can call the analytic higher signature, with an a priori homotopy invariant, the Mishchenko symmetric signature. The former is the generalized index of a certain (generalized) elliptic operator; it plays the role of the index of a family of twisted signature operators in Lusztig’s proof. Recall that the index of a family of operators parameterized by a compact space Y is a certain formal difference of vector bundles over Y , in other words an element of the Grothendieck group of vector bundles, K 0 (Y ).

Applying the conjecture again, f is homotopic to a homeomorphism rel ∂W . Since Whitehead torsion is a topological invariant and τ (M × [0, 1], M × {0}) = 0, τ = 0. Proposition. The Borel Conjecture holds for a group Γ if and only if the algebraic surgery assembly map AΓ : H∗ (BΓ; L• (Z)) → L∗ (Z[Γ]) is an isomorphism and Wh(Γ) = 0, with L∗ = Lh∗ = Ls∗ . Thus if the Borel Conjecture holds for Γ then S∗ (BΓ) = 0, and if (M, ∂M ) is an n-dimensional manifold with boundary such that π1 (M ) = Γ and M is aspherical then S T OP (M rel ∂) = Sn+1 (M ) = Sn+1 (BΓ) = 0 .

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