{"title":"Bridge trisections in ℂℙ2 and the Thom\u0000conjecture","authors":"Peter Lambert-Cole","doi":"10.2140/GT.2020.24.1571","DOIUrl":"https://doi.org/10.2140/GT.2020.24.1571","url":null,"abstract":"In this paper, we develop new techniques for understanding surfaces in $mathbb{CP}^2$ via bridge trisections. Trisections are a novel approach to smooth 4-manifold topology, introduced by Gay and Kirby, that provide an avenue to apply 3-dimensional tools to 4-dimensional problems. Meier and Zupan subsequently developed the theory of bridge trisections for smoothly embedded surfaces in 4-manifolds. The main application of these techniques is a new proof of the Thom conjecture, which posits that algebraic curves in $mathbb{CP}^2$ have minimal genus among all smoothly embedded, oriented surfaces in their homology class. This new proof is notable as it completely avoids any gauge theory or pseudoholomorphic curve techniques.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"25 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75295323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}