Index type invariants for twisted signature complexes and homotopy invariance

TL;DR

This paper introduces a new twisted rho invariant for odd-dimensional manifolds, demonstrating its independence from metric choices and cohomology representatives, and explores its properties, spectral relations, and homotopy invariance.

Contribution

It defines a twisted rho invariant for the odd signature operator, proves its metric and cohomology independence, and establishes homotopy invariance under certain conditions.

Findings

The twisted rho invariant is metric and cohomology independent.

Expressed the twisted eta invariant via spectral flow and eta invariant.

Established a twisted Atiyah-Patodi-Singer theorem for manifolds with boundary.

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 $X$.