Isotopies of surfaces in 4–manifolds via banded unlink diagrams

Hughes, Mark C; Kim, Seungwon; Miller, Maggie

doi:10.2140/gt.2020.24.1519

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Isotopies of surfaces in $4$–manifolds via banded unlink diagrams

Mark C Hughes, Seungwon Kim and Maggie Miller

Geometry & Topology 24 (2020) 1519–1569

DOI: 10.2140/gt.2020.24.1519

Abstract

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 $ℂ P^{2}$ (ie spheres in the generating homology class), proving that many explicit unit surfaces are isotopic to the standard $ℂ P^{1}$ . This strengthens some previously known results about the Gluck twist in $S^{4}$ , related to Kirby problem 4.23.

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Keywords

$4$–manifold, knot, surface, diagram

Mathematical Subject Classification

Primary: 57K45

Secondary: 57K40

References

Publication

Received: 20 February 2019

Revised: 29 March 2020

Accepted: 23 May 2020

Published: 30 September 2020

Proposed: András I Stipsicz

Seconded: Mladen Bestvina, Ciprian Manolescu

Authors

Mark C Hughes
Department of Mathematics Brigham Young University Provo, UT United States
https://math.byu.edu/~hughes/
Seungwon Kim
Center for Geometry and Physics Institute for Basic Science (IBS) Pohang South Korea
https://sites.google.com/view/seungwonkim
Maggie Miller
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States
https://web.math.princeton.edu/~maggiem

Volume 24, issue 3 (2020)