Exotic embedded surfaces and involutions from Real Seiberg–Witten theory

By using Real Seiberg–Witten theory, Miyazawa introduced an invariant of certain [Formula: see text]-manifolds with involution and used this invariant to construct infinitely many exotic involutions on [Formula: see text] and infinitely many exotic smooth embeddings of [Formula: see text] in [Formula: see text]. In this paper, we extend Miyazawa’s construction to a large class of [Formula: see text]-manifolds, giving many infinite families of involutions on [Formula: see text]-manifolds which are conjugate by homeomorphisms but not by diffeomorphisms and many infinite families of exotic embeddings of non-orientable surfaces in [Formula: see text]-manifolds, where exotic means continuously isotopic but not smoothly isotopic. Exoticness of our construction is detected using Real Seiberg–Witten theory. We study Miyazawa’s invariant, relate it to the Real Seiberg–Witten invariants of Tian–Wang and prove various fundamental results concerning the Real Seiberg–Witten invariants such as: relation to positive scalar curvature, wall-crossing, a mod [Formula: see text] formula for spin structures, a localization formula relating ordinary and Real Seiberg–Witten invariants, a connected sum formula and a fiber sum formula.