Involutive Floer Invariants for Closed Four-Manifolds

TL;DR

This paper introduces new involutive Floer invariants for closed four-manifolds, extending existing theories and providing tools to obstruct certain embedded surface configurations.

Contribution

It defines involutive Floer invariants for four-manifolds and demonstrates their application in obstructing pairs of embedded surfaces violating the adjunction inequality.

Findings

Involutive Floer invariants are well-defined for manifolds with $b_2^+ > 4$.

Involutive Seiberg-Witten invariants are well-defined for $b_2^+ > 3$.

These invariants obstruct the existence of disjoint embedded surfaces violating the adjunction inequality.

Abstract

Inspired by the Ozsv\'ath-Szab\'o mixed invariant in ordinary Heegaard Floer theory, we define a mixed invariant $\Phi_{X, \mathfrak{s}}^{I}$ for closed, spin four-manifolds $(X, \mathfrak{s})$ using the cobordism maps on involutive Heegaard Floer homology. The invariant is well-defined whenever $b_{2}^{+}(X) > 4$. We furthermore construct an involutive Seiberg-Witten invariant that is well-defined whenever $b_{2}^{+}(X) > 3$. We show that these involutive invariants obstruct the existence of disjoint pairs of embedded surfaces which both violate the adjunction inequality. As an application, we find that $K3\#(S^2 \times S^2)$ contains no such pair.