An approximation of the $e$-invariant in the stable homotopy category

TL;DR

This paper explores an approximation of the $e$-invariant within the stable homotopy category, providing evidence for a conjecture related to algebraic $K$-theory and index theorems for flat vector bundles.

Contribution

It offers evidence that a certain map from algebraic $K$-theory to a homotopy fiber can be represented by an infinite loop map, advancing understanding of the $e$-invariant and index theory.

Findings

Supports the conjecture that the map can be represented by an infinite loop map

Implications for a refined Bismut-Lott index theorem

Provides a new perspective on the topological index for flat vector bundles

Abstract

In their construction of the topological index for flat vector bundles, Atiyah, Patodi and Singer associate to each flat vector bundle a particular $\mathbb{C/Z}$-$K$-theory class. This assignment determines a map, up to weak homotopy, from $K_{a}\mathbb{C}$, the algebraic $K$-theory space of the complex numbers, to $F_{t,\mathbb{C/Z}}$, the homotopy fiber of the Chern character. In this paper, we give evidence for the conjecture that this map can be represented by an infinite loop map. The result of the paper implies a refined Bismut-Lott index theorem for a compact smooth bundle $E\rightarrow B$ with the fundamental group $\pi_{1}(E,\ast)$ finite for every point $\ast\in E$.