Exotic embedded surfaces and involutions from Real Seiberg–Witten theory

Using Real Seiberg–Witten theory, Miyazawa introduced an invariant of certain 4-manifolds with involution and used this invariant to construct infinitely many exotic involutions on ℂℙ 2 and infinitely many exotic smooth embeddings of ℝℙ 2 in S 4 . In this paper we extend Miyazawa’s construction to a large class of 4-manifolds, giving many infinite families of involutions on 4-manifolds which are conjugate by homeomorphisms but not by diffeomorphisms and many infinite families of exotic embeddings of nonorientable surfaces in 4-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 2 formula for spin structures, a localisation formula relating ordinary and Real Seiberg–Witten invariants, a connected sum formula and a fibre sum formula.