2000/02/24 by Éric Leichtnam, Eric Leichtnam, Leichtnam, Eric +2
Computer Science · Mathematics · #57R20 (primary) #57R67 (secondary) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.GT #msc:57R20 #msc:57R67
paper · pdf · doi:10.48550/arxiv.math/0002201
22 pages
openalex publication_date 2000/02/24 · arxiv created 2000/04/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We extend the notion of the symmetric signature σ(M,r) in Ln(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 ∂ M, where (M,∂ M) → (M,∂ M) is the G-covering associated to r. We need the assumption that C_*(∂ M) ⊗\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 F → F be the associated G-covering. Denote by αm(F) the m-th Novikov-Shubin invariant and by bm(2)(F) the m-th L2-Betti number. We use σ(M,r) to prove the additivity (or cut and paste property) of the higher signatures of Z if we have αm(F) = ∞+ in the case n = 2m and, in the case n = 2m+1, if we have αm(F) = ∞+ and bm(2)(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.