Isotopies of surfaces in 4-manifolds via banded unlink diagrams

Mark C. Hughes; Seungwon Kim; Maggie Miller

doi:10.2140/gt.2020.24.1519

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arXiv · 1804.09169

Isotopies of surfaces in 4-manifolds via banded unlink diagrams

Mark C. Hughes·Seungwon Kim·Maggie Miller

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Abstract

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

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BibTeX RIS

Mark C. Hughes, Seungwon Kim, Maggie Miller. 2019-03-24. Isotopies of surfaces in 4-manifolds via banded unlink diagrams. https://doi.org/10.2140/gt.2020.24.1519

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