{"title":"4流形的多截面","authors":"Gabriel Islambouli, Patrick Naylor","doi":"10.1090/tran/8996","DOIUrl":null,"url":null,"abstract":"We introduce multisections of smooth, closed 4-manifolds, which generalize trisections to decompositions with more than three pieces. This decomposition describes an arbitrary smooth, closed 4-manifold as a sequence of cut systems on a surface. We show how to carry out many smooth cut and paste operations in terms of these cut systems. In particular, we show how to implement a cork twist, whereby we show that an arbitrary exotic pair of smooth 4-manifolds admit 4-sections differing only by one cut system. By carrying out fiber sums and log transforms, we also show that the elliptic fibrations <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E left-parenthesis n right-parenthesis Subscript p comma q\"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">E(n)_{p,q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> all admit genus <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=\"application/x-tex\">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> multisections, and draw explicit diagrams for these manifolds.","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":" 30","pages":"0"},"PeriodicalIF":1.2000,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Multisections of 4-manifolds\",\"authors\":\"Gabriel Islambouli, Patrick Naylor\",\"doi\":\"10.1090/tran/8996\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We introduce multisections of smooth, closed 4-manifolds, which generalize trisections to decompositions with more than three pieces. This decomposition describes an arbitrary smooth, closed 4-manifold as a sequence of cut systems on a surface. We show how to carry out many smooth cut and paste operations in terms of these cut systems. In particular, we show how to implement a cork twist, whereby we show that an arbitrary exotic pair of smooth 4-manifolds admit 4-sections differing only by one cut system. By carrying out fiber sums and log transforms, we also show that the elliptic fibrations <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper E left-parenthesis n right-parenthesis Subscript p comma q\\\"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>n</mml:mi> <mml:msub> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">E(n)_{p,q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> all admit genus <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"3\\\"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=\\\"application/x-tex\\\">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> multisections, and draw explicit diagrams for these manifolds.\",\"PeriodicalId\":23209,\"journal\":{\"name\":\"Transactions of the American Mathematical Society\",\"volume\":\" 30\",\"pages\":\"0\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2023-11-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the American Mathematical Society\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/tran/8996\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/tran/8996","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
We introduce multisections of smooth, closed 4-manifolds, which generalize trisections to decompositions with more than three pieces. This decomposition describes an arbitrary smooth, closed 4-manifold as a sequence of cut systems on a surface. We show how to carry out many smooth cut and paste operations in terms of these cut systems. In particular, we show how to implement a cork twist, whereby we show that an arbitrary exotic pair of smooth 4-manifolds admit 4-sections differing only by one cut system. By carrying out fiber sums and log transforms, we also show that the elliptic fibrations E(n)p,qE(n)_{p,q} all admit genus 33 multisections, and draw explicit diagrams for these manifolds.
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