Volume 24, issue 3 (2020)

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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

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 S4. 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 P2 (ie spheres in the generating homology class), proving that many explicit unit surfaces are isotopic to the standard P1. This strengthens some previously known results about the Gluck twist in S4, related to Kirby problem 4.23.

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$4$–manifold, knot, surface, diagram
Mathematical Subject Classification
Primary: 57K45
Secondary: 57K40
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
Mark C Hughes
Department of Mathematics
Brigham Young University
Provo, UT
United States
Seungwon Kim
Center for Geometry and Physics
Institute for Basic Science (IBS)
South Korea
Maggie Miller
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States