Involutions, links, and Floer cohomologies

We develop a version of Seiberg–Witten Floer cohomology/homotopy type for a ${\mathrm{spin}}^c$ 4-manifold with boundary and with an involution that reverses the ${\mathrm{spin}}^c$ 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.