Involutions, links, and Floer cohomologies

Abstract We develop a version of Seiberg–Witten Floer cohomology/homotopy type for a 4‐manifold with boundary and with an involution that reverses the structure, as well as a version of Floer cohomology/homotopy type for oriented links with nonzero determinant. This framework generalizes the previous work of the authors regarding Floer homotopy type for spin 3‐manifolds with involutions and for knots. Based on this Floer cohomological setting, we prove Frøyshov‐type inequalities that relate topological quantities of 4‐manifolds with certain equivariant homology cobordism invariants. The inequalities and homology cobordism invariants have applications to the topology of unoriented surfaces, the Nielsen realization problem for nonspin 4‐manifolds, and nonsmoothable unoriented surfaces in 4‐manifolds.