Seiberg-Witten Floer K-Theory and Cyclic Group Actions

Given a spin rational homology sphere Y equipped with a cyclic group action preserving the spin structure, I will introduce equivariant refinements of Manolescu's kappa invariant, derived from the equivariant K-theory of the Seiberg--Witten Floer spectrum. These invariants give rise to equivariant relative 10/8-ths type inequalities for equivariant spin cobordisms between rational homology spheres. I will explain how these inequalities provide applications to knot concordance, obstructing cyclic group actions on spin fillings, and genus bounds for knots in punctured 4-manifolds. If time permits I will explain how these invariants are related to equivariant eta-invariants of the Dirac operator, and describe work-in-progress which provides explicit formulas for the S^1-equivariant eta-invariants on Seifert-fibered spaces.