{"title":"树辫群的上同环与外面环","authors":"Jes'us Gonz'alez, Teresa I. Hoekstra-Mendoza","doi":"10.1090/btran/131","DOIUrl":null,"url":null,"abstract":"<p>For a tree <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> and a positive integer <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>, let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B Subscript n Baseline upper T\">\n <mml:semantics>\n <mml:mrow>\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:mrow>\n <mml:annotation encoding=\"application/x-tex\">B_nT</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> denote the <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>-strand braid group on <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 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":"{\"title\":\"Cohomology ring of tree braid groups and exterior face rings\",\"authors\":\"Jes'us Gonz'alez, Teresa I. Hoekstra-Mendoza\",\"doi\":\"10.1090/btran/131\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For a tree <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> and a positive integer <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>, let <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper B Subscript n Baseline upper T\\\">\\n <mml:semantics>\\n <mml:mrow>\\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:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">B_nT</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> denote the <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>-strand braid group on <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 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}","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
摘要
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。
Cohomology ring of tree braid groups and exterior face rings
For a tree TT and a positive integer nn, let BnTB_nT denote the nn-strand braid group on TT. We use discrete Morse theory techniques to show that the cohomology ring H∗(BnT)H^*(B_nT) is encoded by an explicit abstract simplicial complex KnTK_nT that measures nn-local interactions among essential vertices of TT. We show that, in many cases (for instance when TT is a binary tree), H∗(BnT)H^*(B_nT) is the exterior face ring determined by KnTK_nT.
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