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Relative genus bounds in indefinite four-manifolds

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Abstract

Given a closed four-manifold X with an indefinite intersection form, we consider smoothly embedded surfaces in \(X {\setminus } \smash {\mathring{B}^4}\), with boundary a knot \(K \subset S^3\). We give several methods to bound the genus of such surfaces in a fixed homology class. Our tools include adjunction inequalities and the \(10/8 + 4\) theorem. In particular, we present obstructions to a knot being H-slice (that is, bounding a null-homologous disk) in a four-manifold and show that the set of H-slice knots can detect exotic smooth structures on closed 4-manifolds.

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Acknowledgements

We are grateful to Matt Hedden, Michael Klug, Maggie Miller, Benjamin Ruppik, and Ian Zemke for helpful conversations. We also thank Gordana Matić, Anubhav Mukherjee, Kouichi Yasui, and the referee for comments on a previous version of the paper.

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  1. Department of Mathematics, Stanford University, 450 Jane Stanford Way, Stanford, CA, 94305, USA

    Ciprian Manolescu

  2. HUN-REN Alfréd Rényi Institute of Mathematics, Réaltanoda utca 13-15, 1053, Budapest, Hungary

    Marco Marengon

  3. Department of Mathematics, University of Texas at Austin, 2515, Speedway, Austin, TX, 78712, USA

    Lisa Piccirillo

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  1. Ciprian Manolescu
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Correspondence to Ciprian Manolescu.

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CM was supported by NSF Grant DMS-2003488 and a Simons Investigator Award. MM was supported by NSF FRG Grant DMS-1563615 and the Max Planck Institute for Mathematics. LP was supported by NSF postdoctoral fellowship DMS-1902735 and the Max Planck Institute for Mathematics.

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Manolescu, C., Marengon, M. & Piccirillo, L. Relative genus bounds in indefinite four-manifolds. Math. Ann. (2024). https://doi.org/10.1007/s00208-023-02787-4

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