Comparing Periodic Point Invariants for Parameterized Families of Maps

TL;DR

This paper demonstrates that for parameterized families of maps, the fiberwise Fuller trace is a more sensitive invariant than the collection of Reidemeister traces, resolving a conjecture and highlighting differences from the single-map case.

Contribution

It establishes that the fiberwise Fuller trace is strictly more sensitive than Reidemeister traces for families of maps, contrasting with the single-map scenario.

Findings

Fiberwise Fuller trace is more sensitive than Reidemeister traces for families of maps.

Resolves a conjecture of Malkiewich and Ponto.

Highlights differences between single-map and parameterized family invariants.

Abstract

We compare different periodic point invariants for families of maps parameterized over a compact manifold. Malkiewich and Ponto showed that, in the case of a single map, the Fuller trace is equivalent to the collection of Reidmeister traces of iterates. In this paper, we show that, in contrast to the case of a single map, the fiberwise Fuller trace is a strictly more sensitive invariant than the collection of fiberwise Reidemeister traces of iterates. This resolves a conjecture of Malkiewich and Ponto.