On the cut-and-paste property of higher signatures of a closed oriented manifold

We extend the notion of the symmetric signature σ(M̄,r)∈Ln(R) for a compact n-dimensional manifold M without boundary, a reference map r : M→BG and a homomorphism of rings with involutions β : ZG→R to the case with boundary ∂M, where (M̄, ∂M)→(M, ∂M) is the G-covering associated to r. We need the assumption that C∗(∂M) ⊗ZG R is R-chain homotopy equivalent to a R-chain complex D∗ with trivial mth differential for n=2m resp. n=2m+1. We prove a glueing formula, homotopy invariance and additivity for this new notion. Let Z be a closed oriented manifold with reference map Z→BG. Let F⊂Z be a cutting codimension one submanifold F⊂Z and let F̄→F be the associated G-covering. Denote by αm(F̄) the mth Novikov–Shubin invariant and by bm(2)(F̄) the mth L2-Betti number. If for the discrete group G the Baum–Connes assembly map is rationally injective, then we use σ(M̄,r) to prove the additivity (or cut and paste property) of the higher signatures of Z, if we have αm(F̄)=∞+ in the case n=2m and, in the case n=2m+1, if we have αm(F̄)=∞+ and bm(2)(F̄)=0. This additivity result had been proved (by a different method) in (On the Homotopy Invariance of Higher Signatures for Mainfolds with Boundary, preprint, 1999, Corollary 0.4) when G is Gromov hyperbolic or virtually nilpotent. We give new examples, where these conditions are not satisfied and additivity fails. lain at the end of the introduction why our paper is greatly motivated by and partially extends some of the work of Leichtnam et al. (On the Homotopy Invariance of Higher Signatures for Mainfolds with Boundary, preprint, 1999), Lott (Math. Ann., 1999) and Weinberger (Contemporary Mathematics, 1999, p. 231).