New invariants of involutions from Seiberg-Witten Floer theory

Abstract

We study equivariant Seiberg-Witten Floer theory of rational homology $3$-spheres in the special case where the group action is given by an involution. The case of involutions deserves special attention because we can couple the involution to the charge conjugation symmetry of Seiberg-Witten theory. This leads to new Floer-theoretic invariants which we study and apply in a variety of applications. In particular, we construct a series of delta-invariants $\delta^E_*, \delta^R_*, \delta^S_*$ which are the equivariant equivalents of the Ozsv\'ath-Szab\'o $d$-invariant. The delta-invariants come in three types: equivariant, Real and spin depending on the type of the spin$^c$-structure involved. The delta-invariants satisfy many useful properties, including a Froyshov-type inequality for equivariant cobordisms. We compute the delta-invariants in a wide range of examples including: equivariant plumbings, branched double covers of knots and equivariant Dehn surgery. We also consider various applications including obstructions to extending involutions over bounding $4$-manifolds, non-smoothable involutions on $4$-manifolds with boundary, equivariant embeddings of $3$-manifolds in $4$-manifolds and non-orientable surfaces bounding knots.