A survey of the homology cobordism group

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS

ACS Applied Bio Materials 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":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bull/1806","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","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

ACS Applied Bio Materials Chemistry-Chemistry (all)

CiteScore

9.40

自引率

2.10%

发文量

464