An adjunction inequality for Real embedded surfaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911485295329280 |
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| author | Baraglia, David |
| author_facet | Baraglia, David |
| contents | A Real structure on a $4$-manifold $X$ is an orientation preserving smooth involution $σ$. We say that an embedded surface $Σ\subset X$ is Real if $σ$ maps $Σ$ to itself orientation reversingly. We prove that a cohomology class $u \in H^2(X ; \mathbb{Z})$ can be represented by a Real embedded surface if and only if $u$ can be lifted to a class in equivariant cohomology $H^2_{\mathbb{Z}_2}(X ; \mathbb{Z}_-)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05667 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An adjunction inequality for Real embedded surfaces Baraglia, David Geometric Topology Differential Geometry A Real structure on a $4$-manifold $X$ is an orientation preserving smooth involution $σ$. We say that an embedded surface $Σ\subset X$ is Real if $σ$ maps $Σ$ to itself orientation reversingly. We prove that a cohomology class $u \in H^2(X ; \mathbb{Z})$ can be represented by a Real embedded surface if and only if $u$ can be lifted to a class in equivariant cohomology $H^2_{\mathbb{Z}_2}(X ; \mathbb{Z}_-)$. 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. |
| title | An adjunction inequality for Real embedded surfaces |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/2507.05667 |