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Type: Journal article
Title: Index type invariants for twisted signature complexes and homotopy invariance
Author: Benameur, M.
Mathai, V.
Citation: Mathematical Proceedings of the Cambridge Philosophical Society, 2014; 156(3):473-503
Publisher: Cambridge University Press
Issue Date: 2014
ISSN: 0305-0041
Statement of
Moulay Tahar Benameur, Varghese Mathai
Abstract: For a closed, oriented, odd dimensional manifold X, we define the rho invariant ρ(X,Ɛ,H) for the twisted odd signature operator valued in a flat hermitian vector bundle , where H = ∑ i j+1 H 2j+1 is an odd-degree closed differential form on X and H 2j+1 is a real-valued differential form of degree 2j+1. We show that ρ(X,Ɛ,H) is independent of the choice of metrics on X and Ɛ of the representative H in the cohomology class [H]. We establish some basic functorial properties of the twisted rho invariant. We express the twisted eta invariant in terms of spectral flow and the usual eta invariant. In particular, we get a simple expression for it on closed oriented 3-dimensional manifolds with a degree three flux form. A core technique used is our analogue of the Atiyah–Patodi–Singer theorem, which we establish for the twisted signature operator on a compact, oriented manifold with boundary. The homotopy invariance of the rho invariant ρ(X,Ɛ,H) more delicate to establish, and is settled under further hypotheses on the fundamental group of X.
Rights: © Cambridge Philosophical Society 2014
RMID: 0030023002
DOI: 10.1017/S030500411400005X
Appears in Collections:Mathematical Sciences publications

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