On the Fr{\o}yshov invariant and monopole Lefschetz number

TL;DR

This paper derives a formula for the Lefschetz number of an involution on a rational homology 3-sphere using monopole Floer homology, connecting gauge theory, knot theory, and 4-manifold invariants.

Contribution

It introduces a skein-theoretic approach to compute the Lefschetz number in monopole Floer homology, linking it to the Murasugi signature and Fr{ }yshov invariants.

Findings

Derived a Lefschetz number formula involving Murasugi signature and Fr{ }yshov invariants.

Applied the formula to gauge theory, knot theory, and contact geometry.

Connected involution actions on 3-manifolds to 4-manifold invariants.

Abstract

Given an involution on a rational homology 3-sphere $Y$ with quotient the $3$-sphere, we prove a formula for the Lefschetz number of the map induced by this involution in the reduced monopole Floer homology. This formula is motivated by a variant of Witten's conjecture relating the Donaldson and Seiberg--Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic argument, making use of an exact triangle in monopole Floer homology, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Fr{\o}yshov invariants associated to spin structures on $Y$. We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology.