Involutive Floer Invariants for Closed Four-Manifolds

Inspired by the Ozsváth-Szabó mixed invariant in ordinary Heegaard Floer theory, we define a mixed invariant ${\mathrm{\Phi }}_{X,\mathfrak{s}}^I$ 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 $K3\mathrm{\#}(S^2\times S^2)$ contains no such pair.