Coincidence invariants and higher Reidemeister traces

TL;DR

This paper explores the relationship between algebraic and topological coincidence invariants, extending the connection between Lefschetz number and index to coincidences, and discusses challenges in generalizing to Reidemeister traces.

Contribution

It demonstrates that the identification of Lefschetz number and index via symmetric monoidal trace extends to coincidence invariants, highlighting difficulties in generalizing to Reidemeister traces.

Findings

Extension of Lefschetz number and index identification to coincidences

Formal properties of symmetric monoidal trace underpin these invariants

Identifies challenges in generalizing to coincidence Reidemeister trace

Abstract

The Lefschetz number and fixed point index can be thought of as two different descriptions of the same invariant. The Lefschetz number is algebraic and defined using homology. The index is defined more directly from the topology and is a stable homotopy class. Both the Lefschetz number and index admit generalizations to coincidences and the comparison of these invariants retains its central role. In this paper we show that the identification of the Lefschetz number and index using formal properties of the symmetric monoidal trace extends to coincidence invariants. This perspective on the coincidence index and Lefschetz number also suggests difficulties for generalizations to a coincidence Reidemeister trace.