A survey of the homology cobordism group

IF 2 3区数学 Q1 MATHEMATICS

Bulletin of the American Mathematical Society Pub Date : 2023-10-06 DOI:10.1090/bull/1806

Oğuz Şavk

{"title":"A survey of the homology cobordism group","authors":"Oğuz Şavk","doi":"10.1090/bull/1806","DOIUrl":null,"url":null,"abstract":"In this survey, we present the most recent highlights from the study of the homology cobordism group, with particular emphasis on its long-standing and rich history in the context of smooth manifolds. Further, we list various results on its algebraic structure and discuss its crucial role in the development of low-dimensional topology. Also, we share a series of open problems about the behavior of homology <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>-spheres and the structure of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Theta Subscript double-struck upper Z Superscript 3\"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant=\"normal\">Θ</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>3</mml:mn> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">\\Theta _{\\mathbb {Z}}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Finally, we briefly discuss the knot concordance group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper C\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the rational homology cobordism group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Theta Subscript double-struck upper Q Superscript 3\"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant=\"normal\">Θ</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>3</mml:mn> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">\\Theta _{\\mathbb {Q}}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, focusing on their algebraic structures, relating them to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Theta Subscript double-struck upper Z Superscript 3\"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant=\"normal\">Θ</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>3</mml:mn> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">\\Theta _{\\mathbb {Z}}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and highlighting several open problems. The appendix is a compilation of several constructions and presentations of homology <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>-spheres introduced by Brieskorn, Dehn, Gordon, Seifert, Siebenmann, and Waldhausen.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":"105 1","pages":"0"},"PeriodicalIF":2.0000,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the American Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bull/1806","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

In this survey, we present the most recent highlights from the study of the homology cobordism group, with particular emphasis on its long-standing and rich history in the context of smooth manifolds. Further, we list various results on its algebraic structure and discuss its crucial role in the development of low-dimensional topology. Also, we share a series of open problems about the behavior of homology 3 3 -spheres and the structure of Θ Z 3 \Theta _{\mathbb {Z}}^3 . Finally, we briefly discuss the knot concordance group C \mathcal {C} and the rational homology cobordism group Θ Q 3 \Theta _{\mathbb {Q}}^3 , focusing on their algebraic structures, relating them to Θ Z 3 \Theta _{\mathbb {Z}}^3 , and highlighting several open problems. The appendix is a compilation of several constructions and presentations of homology 3 3 -spheres introduced by Brieskorn, Dehn, Gordon, Seifert, Siebenmann, and Waldhausen.

查看原文本刊更多论文

同源配位群的综述

在这个调查中，我们提出了最近的亮点，从研究的同调配群，特别强调其长期和丰富的历史，在光滑流形的背景下。进一步，我们列出了关于它的代数结构的各种结果，并讨论了它在低维拓扑发展中的重要作用。此外，我们还讨论了一系列关于同调33 -球的行为和Θ z3 \Theta _{\mathbb {Z}}^3结构的开放问题。最后，我们简要讨论了结谐和群C \mathcal {C}和有理同调群Θ Q 3 \Theta _{\mathbb {Q}}^3，重点讨论了它们的代数结构，并将它们与Θ Z 3 \Theta _{\mathbb {Z}}^3联系起来，突出了几个开放问题。附录是由Brieskorn, Dehn, Gordon, Seifert, Siebenmann和Waldhausen介绍的几个同构33球的构造和表示的汇编。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Bulletin of the American Mathematical Society 数学-数学

CiteScore

2.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.