On localized signature and higher rho invariant of fibered manifolds

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).