On localized signature and higher rho invariant of fibered manifolds
Hongzhi Liu
Shanghai University of Finance and Economics, ChinaJinmin Wang
Fudan University, Shanghai, China
![On localized signature and higher rho invariant of fibered manifolds cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-jncg-volume-15-issue-3.png&w=3840&q=90)
Abstract
The higher index of the signature operator is a far-reaching generalization of the 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 the higher index and the higher rho invariant of the signature operator on a fibered manifold. Our result implies the classical product formula for the numerical signature of a fibered manifold obtained by Chern, Hirzebruch, and Serre (1957). We also give a new proof of the product formula for the higher rho invariant of the signature operator on a product manifold, which is parallel to the product formula for the higher rho invariant of Dirac operator on a product manifold obtained by Xie and Yu (2014) and Zeidler (2016).
Cite this article
Hongzhi Liu, Jinmin Wang, On localized signature and higher rho invariant of fibered manifolds. J. Noncommut. Geom. 15 (2021), no. 3, pp. 919–949
DOI 10.4171/JNCG/426