{"title":"Diagonal complexes for surfaces of finite type and surfaces with involution","authors":"G. Panina, J. Gordon","doi":"10.1090/spmj/1709","DOIUrl":null,"url":null,"abstract":"<p>Two constructions are studied that are inspired by the ideas of a recent paper by the authors.</p>\n\n<p>— The <italic> diagonal complex</italic> <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper D\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {D}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and its barycentric subdivision <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper B script upper D\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">B</mml:mi>\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {BD}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> related to an oriented surface of finite type <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\n <mml:semantics>\n <mml:mi>F</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> equipped with a number of labeled marked points. This time, unlike the paper mentioned above, boundary components without marked points are allowed, called <italic>holes</italic>.</p>\n\n<p>— The <italic>symmetric diagonal complex</italic> <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper D Superscript i n v\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>inv</mml:mi>\n </mml:mrow>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {D}^{\\operatorname {inv}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and its barycentric subdivision <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper B script upper D Superscript i n v\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">B</mml:mi>\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>inv</mml:mi>\n </mml:mrow>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {BD}^{\\operatorname {inv}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> related to a <italic>symmetric</italic> (=with an involution) oriented surface <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\n <mml:semantics>\n <mml:mi>F</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> equipped with a number of (symmetrically placed) labeled marked points.</p>\n\n<p>The symmetric complex is shown to be homotopy equivalent to the complex of a surface obtained by “taking a half” of the initial symmetric surface.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"St Petersburg Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1709","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Two constructions are studied that are inspired by the ideas of a recent paper by the authors.
— The diagonal complexD\mathcal {D} and its barycentric subdivision BD\mathcal {BD} related to an oriented surface of finite type FF equipped with a number of labeled marked points. This time, unlike the paper mentioned above, boundary components without marked points are allowed, called holes.
— The symmetric diagonal complexDinv\mathcal {D}^{\operatorname {inv}} and its barycentric subdivision BDinv\mathcal {BD}^{\operatorname {inv}} related to a symmetric (=with an involution) oriented surface FF equipped with a number of (symmetrically placed) labeled marked points.
The symmetric complex is shown to be homotopy equivalent to the complex of a surface obtained by “taking a half” of the initial symmetric surface.
期刊介绍：
This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.
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