{"title":"Isotopies of surfaces in 4–manifolds via banded\nunlink diagrams","authors":"M. Hughes, Seungwon Kim, Maggie Miller","doi":"10.2140/GT.2020.24.1519","DOIUrl":null,"url":null,"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.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"10 1","pages":""},"PeriodicalIF":2.0000,"publicationDate":"2018-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"25","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometry & Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/GT.2020.24.1519","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 25
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.
期刊介绍:
Geometry and Topology is a fully refereed journal covering all of geometry and topology, broadly understood. G&T is published in electronic and print formats by Mathematical Sciences Publishers.
The purpose of Geometry & Topology is the advancement of mathematics. Editors evaluate submitted papers strictly on the basis of scientific merit, without regard to authors" nationality, country of residence, institutional affiliation, sex, ethnic origin, or political views.