Relative fixed point theory

TL;DR

This paper develops a generalized fixed point theory using bicategorical traces to define relative Lefschetz numbers and Reidemeister traces, proving a relative Lefschetz fixed point theorem and its converse.

Contribution

It introduces a new framework for relative fixed point invariants using bicategory traces, extending classical Lefschetz theory.

Findings

Defined relative Lefschetz numbers and Reidemeister traces using bicategorical traces.

Proved a relative Lefschetz fixed point theorem and its converse.

Unified various forms of fixed point invariants through functorial trace methods.

Abstract

The Lefschetz fixed point theorem and its converse have many generalizations. One of these generalizations is to endomorphisms of a space relative to a fixed subspace. In this paper we define relative Lefschetz numbers and Reidemeister traces using traces in bicategories with shadows. We use the functoriality of this trace to identify different forms of these invariants and to prove a relative Lefschetz fixed point theorem and its converse.