Index type invariants for twisted signature complexes and homotopy invariance

Abstract

For a closed, oriented, odd dimensional manifold $X$, we define the rho invariant $\rho (X,E,H)$ for the twisted odd signature operator valued in a flat hermitian vector bundle $E$, where $H=\sum i^{j+1}{H}_{2j+1}$ 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 the twisted rho invariant $\rho (X,E,H)$ is independent of the choice of metrics on $X$ and $E$ and 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 $\rho (X,E,H)$ is more delicate to establish, and is settled under further hypotheses on the fundamental group of $X$.

33 pages, to appear in, Math. Proc. Cambridge Philos. Soc

References in corpus (6)

• Generalized complex geometry
• Index, eta and rho-invariants on foliated bundles
• Analytic Torsion of Z_2-graded Elliptic Complexes
• Twisted Analytic Torsion
• Spectral sections, twisted rho invariants and positive scalar curvature
• Conformal invariants of twisted Dirac operators and positive scalar curvature

Cited by in corpus (5)

• Spectral sections, twisted rho invariants and positive scalar curvature
• Conformal invariants of twisted Dirac operators and positive scalar curvature
• The Higson-Roe exact sequence and ${\ell }^2$ eta invariants
• Positive scalar curvature via end-periodic manifolds
• Signature, Toledo invariant and surface group representations in the real symplectic group