Seiberg-Witten Floer K-theory and cyclic group actions on spin four-manifolds with boundary

TL;DR

This paper develops equivariant Seiberg-Witten Floer K-theory invariants for spin rational homology spheres with cyclic group actions, deriving inequalities and applications in knot theory and 4-manifold topology.

Contribution

It introduces equivariant refinements of Seiberg-Witten invariants incorporating cyclic group actions, leading to new inequalities and topological obstructions.

Findings

Derived equivariant 10/8-ths inequalities for spin cobordisms.

Provided obstructions to extending cyclic actions to spin fillings.

Established genus bounds for knots in complex 4-manifolds.

Abstract

Given a spin rational homology sphere $Y$ equipped with a $\mathbb{Z}/m$-action preserving the spin structure, we use the Seiberg--Witten equations to define equivariant refinements of the invariant $\kappa(Y)$ from \cite{Man14}, which take the form of a finite subset of elements in a lattice constructed from the representation ring of a twisted product of $\text{Pin}(2)$ and $\mathbb{Z}/m$. The main theorems consist of equivariant relative 10/8-ths type inequalities for spin equivariant cobordisms between rational homology spheres. We provide applications to knot concordance, give obstructions to extending cyclic group actions to spin fillings, and via taking branched covers we obtain genus bounds for knots in punctured 4-manifolds. In some cases, these bounds are strong enough to determine the relative genus for a large class of knots within certain homology classes in $\mathbb{C} P^{2}\#\mathbb{C} P^{2}$, $S^{2}\times S^{2}\# S^{2}\times S^{2}$, $\mathbb{C} P^{2}\# S^{2}\times S^{2}$, and homotopy $K3$ surfaces.