Isotopies of surfaces in 4–manifolds via banded unlink diagrams

Hughes, Mark C¹ ; Kim, Seungwon² ; Miller, Maggie³

¹ Department of Mathematics, Brigham Young University, Provo, UT, United States
² Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, South Korea
³ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, United States

Geometry & topology, Tome 24 (2020) no. 3, pp. 1519-1569.

Résumé

We study surfaces embedded in $4$–manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary $4$–manifold. This extends work of Swenton and Kearton–Kurlin in $S^4$. As an application, we show that bridge trisections of isotopic surfaces in a trisected $4$–manifold are related by a sequence of perturbations and deperturbations, affirmatively proving a conjecture of Meier and Zupan. We also exhibit several isotopies of unit surfaces in $ℂP^2$ (ie spheres in the generating homology class), proving that many explicit unit surfaces are isotopic to the standard $ℂP^1$. This strengthens some previously known results about the Gluck twist in $S^4$, related to Kirby problem 4.23.

Détail

DOI : 10.2140/gt.2020.24.1519

Keywords: $4$–manifold, knot, surface, diagram

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