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

Abstract

We extend the notion of the symmetric signature $\sigma(\bar{M},r)$ in L^n(R) for a compact n-dimensional manifold M without boundary, a reference map r from M to BG and a homomorphism of rings with involutions from ZG to R to the case with boundary $\partial M$, where $(\bar{M},\bar{\partial M}) \to (M,\partial M)$ is the G-covering associated to r. We need the assumption that $C_*(\bar{\partial M}) \otimes_{\zz G} R$ isR-chain homotopy equivalent to a R-chain complex D_* with trivial m-th differential for n = 2m resp. n = 2m+1. Let Z be a closed oriented manifold with reference map BG. Let F be a cutting codimension one submanifold in Z and let $\bar{F} \to F$ be the associated $G$-covering. Denote by $\alpha_m(\bar{F})$ the m-th Novikov-Shubin invariant and by $b_m^{(2)}(\bar{F})$ the m-th L^2-Betti number. We use $\sigma(\bar{M},r)$ to prove the additivity (or cut and paste property) of the higher signatures of Z if we have $\alpha_m(\bar{F}) = \infty^+$ in the case n = 2m and, in the case n = 2m+1, if we have $\alpha_m(\bar{F}) = \infty^+$ and $b_m^{(2)}(\bar{F}) = 0$. We give examples, where these conditions are not satisfied and additivity fails. Our work is motivated by the one of Leichtnam-Lott-Piazza, Lott and Weinberger.