Cohomology ring of tree braid groups and exterior face rings

Transactions of the American Mathematical Society, Series B Pub Date : 2021-06-24 DOI:10.1090/btran/131

Jes'us Gonz'alez, Teresa I. Hoekstra-Mendoza

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We use discrete Morse theory techniques to show that the cohomology ring <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript asterisk Baseline left-parenthesis upper B Subscript n Baseline upper T right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mi>H</mml:mi>\n <mml:mo>∗</mml:mo>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>B</mml:mi>\n <mml:mi>n</mml:mi>\n </mml:msub>\n <mml:mi>T</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">H^*(B_nT)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is encoded by an explicit abstract simplicial complex <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K Subscript n Baseline upper T\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>K</mml:mi>\n <mml:mi>n</mml:mi>\n </mml:msub>\n <mml:mi>T</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">K_nT</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> that measures <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\n <mml:semantics>\n <mml:mi>n</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-local interactions among essential vertices of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T\">\n <mml:semantics>\n <mml:mi>T</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">T</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We show that, in many cases (for instance when <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T\">\n <mml:semantics>\n <mml:mi>T</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">T</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a binary tree), <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript asterisk Baseline left-parenthesis upper B Subscript n Baseline upper T right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mi>H</mml:mi>\n <mml:mo>∗</mml:mo>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>B</mml:mi>\n <mml:mi>n</mml:mi>\n </mml:msub>\n <mml:mi>T</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">H^*(B_nT)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is the exterior face ring determined by <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K Subscript n Baseline upper T\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>K</mml:mi>\n <mml:mi>n</mml:mi>\n </mml:msub>\n <mml:mi>T</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">K_nT</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"67 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/btran/131","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 4

Abstract

For a tree T T and a positive integer n n , let B n T B_nT denote the n n -strand braid group on T T . We use discrete Morse theory techniques to show that the cohomology ring H ∗ ( B n T ) H^*(B_nT) is encoded by an explicit abstract simplicial complex K n T K_nT that measures n n -local interactions among essential vertices of T T . We show that, in many cases (for instance when T T is a binary tree), H ∗ ( B n T ) H^*(B_nT) is the exterior face ring determined by K n T K_nT .

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树辫群的上同环与外面环

对于一棵树T和一个积极的妥协n n n，让我们不要把braid小组放在T T上。我们用discrete莫尔斯理论techniques展示那个《cohomology环 H∗ ( B n T ) H ^ * (B_nT)是encoded by an explicit抽象simplicial情结 K n T K_nT那措施n n -local interactions》essential vertices of T T。我们在许多案子那个节目,(例如当T T是一个二进制树的 ), H∗ ( B n T ) H ^ * (B_nT)是《拳台“淡入脸intended: K n T K_nT。

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Transactions of the American Mathematical Society, Series B

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1.70

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0.00%

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