Equivalence of different definitions of higher $\rho$ invariants

TL;DR

This paper proves that two different definitions of higher rho invariants, associated with orientation-preserving homotopy equivalences between closed oriented smooth manifolds, are mathematically equivalent.

Contribution

It establishes the equivalence of two main approaches to defining higher rho invariants in the context of smooth manifolds.

Findings

The two definitions of higher rho invariants are shown to be equivalent.

Provides a unified understanding of higher rho invariants in topology.

Strengthens the theoretical foundation for invariants in geometric topology.

Abstract

For each orientation-preserving homotopy equivalence between two closed oriented smooth manifolds, there are mainly two different approaches to the higher $\rho$ invariant associated to this homotopy equivalence. In this article, we show that these two definitions of the higher $\rho$ invariant are equivalent.