Exotic embedded surfaces and involutions from Real Seiberg-Witten theory

2025/03/31 by Baraglia, David · 3 citations
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2504.00281

Abstract

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 \mathbbCP² and infinitely many exotic smooth embeddings of \mathbbRP² in S⁴. 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 non-orientable 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.

Cited by

• A Fourier-Mukai Transform For KR Theory
• A note on real Heegaard Floer homology and localization
• An adjunction inequality for Real embedded surfaces

Related

• From diffeomorphisms to exotic phenomena in small 4-manifolds
• Constraints on embedded spheres and real projective planes in 4-manifolds from Seiberg-Witten theory
• A gauge theoretic invariant of embedded surfaces in 4-manifolds and exotic P²-knots
• Family Bauer--Furuta invariant, Exotic Surfaces and Smale conjecture
• Family Seiberg-Witten invariants and wall crossing formulas