An adjunction inequality for Real embedded surfaces

2025/07/08 by Baraglia, David
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2507.05667

Abstract

A Real structure on a 4-manifold X is an orientation preserving smooth involution σ. We say that an embedded surface Σ⊂ X is Real if σ maps Σ to itself orientation reversingly. We prove that a cohomology class u ∈ H²(X ; ℤ) can be represented by a Real embedded surface if and only if u can be lifted to a class in equivariant cohomology H²_ℤ2(X ; ℤ_-). We prove that if the Real Seiberg--Witten invariants of X are non-zero then the genus of Real embedded surfaces in X satisfy an adjunction inequality. We prove two versions of the adjunction inequality, one for non-negative self-intersection and one for arbitrary self-intersection. We show with examples that the minimal genus of Real embedded surfaces can be larger than the minimal genus of arbitrary embedded surfaces.

Citations

• A satellite formula for real Seiberg-Witten Floer homotopy types
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• Higher Type Adjunction Inequalities in Seiberg-Witten Theory
• Higher type adjunction inequalities for Donaldson invariants
• Relative Seiberg-Witten and Ozsvath-Szabo 4-dimensional invariants with respect to embedded surfaces