On the topological contents of eta invariants

TL;DR

This paper explores a universal bordism invariant derived from the eta-invariant, connecting analytic and homotopy theoretic perspectives, and unifies classical invariants like the Adams e-invariant and rho-invariants.

Contribution

It introduces a secondary index theorem establishing the equivalence of analytic and topological constructions of the invariant, providing intrinsic formulas for bordism invariants.

Findings

Proves a secondary index theorem linking analytic and topological eta-invariants.

Derives classical invariants as special cases of the universal bordism invariant.

Provides explicit intrinsic expressions for these invariants.

Abstract

We discuss an universal bordism invariant obtained from the Atiyah-Patodi-Singer eta-invariant from the analytic and homotopy theoretic point of view. Classical invariants like the Adams e-invariant, $\rho$-invariants and $String$-bordism invariants are derived as special cases. The main results are a secondary index theorem about the coincidence of the analytic and topological constructions and intrinsic expressions for the bordism invariants.