Topological $K$-theory with coefficients and the $e$-invariant

TL;DR

This paper compares different invariants of flat vector bundles, explores their homotopy properties, and relates them to the $e$-invariant, providing new insights into their geometric and algebraic structures.

Contribution

It demonstrates the equivalence of invariants from different sources, analyzes the $e$-invariant's properties, and recovers known formulas for Seifert homology spheres, proposing a new conjecture about its infinite loop map representation.

Findings

Invariants from Atiyah et al. and Jones et al. induce the same map $e$.

The map $e$ relates to the homotopy fiber of the Chern character.

Recovered the formula for the real part of $e$-invariants of Seifert homology spheres.

Abstract

We compare the invariants of flat vector bundles defined by Atiyah et al. and Jones et al. and prove that, up to weak homotopy, they induce the same map, denoted by $e$, from the $0$-connective algebraic $K$-theory space of the complex numbers to the homotopy fiber of the Chern character. We examine homotopy properties of this map and its relation with other known invariants. In addition, using the formula for $\tilde{\xi}$-invariants of lens spaces derived from Donnelly's fixed point theorem and the $4$-dimensional cobordisms constructed via relative Kirby diagrams, we recover the formula for the real part of $e$-invariants of Seifert homology spheres given by Jones and Westbury, up to sign. We conjecture that this geometrically defined map $e$ can be represented by an infinite loop map. The results in its companion paper [Wang2] give strong evidence for this conjecture.