The linearity of fixed point invariants

TL;DR

This paper establishes general decomposition theorems for fixed-point invariants like the Lefschetz number and Reidemeister trace, using formal bicategorical trace methods, enabling broader generalizations.

Contribution

It introduces formal, general decomposition theorems for fixed-point invariants within a bicategorical framework, facilitating extensions to other contexts.

Findings

Proves additivity of Lefschetz number and Reidemeister trace

Uses formal bicategorical trace methods for proofs

Enables generalization to equivariant and fiberwise homotopy theory

Abstract

We prove two general decomposition theorems for fixed-point invariants: one for the Lefschetz number and one for the Reidemeister trace. These theorems imply the familiar additivity results for these invariants. Moreover, the proofs of these theorems are essentially formal, taking place in the abstract context of bicategorical traces. This makes it straightforward to generalize the theory to analogous invariants in other contexts, such as equivariant and fiberwise homotopy theory.