On localized signature and higher rho invariant of fibered manifolds

TL;DR

This paper establishes product formulas for higher index and rho invariants of signature operators on fibered manifolds, extending classical results and providing new proofs for product formulas in higher rho invariants.

Contribution

It proves product formulas for higher index and rho invariants of signature operators on fibered manifolds, generalizing classical signature product formulas and offering new proofs.

Findings

Product formulas for higher index of signature operator on fibered manifolds.

Product formulas for higher rho invariant of signature operator on fibered manifolds.

Extension of classical signature product formula to higher invariants.

Abstract

Higher index of signature operator is a far reaching generalization of signature of a closed oriented manifold. When two closed oriented manifolds are homotopy equivalent, one can define a secondary invariant of the relative signature operator called higher rho invariant. The higher rho invariant detects the topological nonrigidity of a manifold. In this paper, we prove product formulas for higher index and higher rho invariant of signature operator on fibered manifolds. Our result implies the classical product formula for numerical signature of fiber manifolds obtained by Chern, Hirzebruch, and Serre in "On the index of a fibered manifold". We also give a new proof of the product formula for higher rho invariant of signature operator on product manifolds, which is parallel to the product formula for higher rho invariant of Dirac operator on product manifolds obtained by Xie and Yu in "Positive scalar curvature, higher rho invariants and localization algebras" and Zeidler in "Positive scalar curvature and product formulas for secondary index invariants".