{"title":"通过有效同源性和离散向量场计算普遍盖的同源性","authors":"Miguel Angel Marco-Buzunariz, Ana Romero","doi":"arxiv-2409.06357","DOIUrl":null,"url":null,"abstract":"Effective homology techniques allow us to compute homology groups of a wide\nfamily of topological spaces. By the Whitehead tower method, this can also be\nused to compute higher homotopy groups. However, some of these techniques (in\nparticular, the Whitehead tower) rely on the assumption that the starting space\nis simply connected. For some applications, this problem could be circumvented\nby replacing the space by its universal cover, which is a simply connected\nspace that shares the higher homotopy groups of the initial space. In this\npaper, we formalize a simplicial construction for the universal cover, and\nrepresent it as a twisted cartesian product. As we show with some examples, the universal cover of a space with effective\nhomology does not necessarily have effective homology in general. We show two\nindependent sufficient conditions that can ensure it: one is based on a\nnilpotency property of the fundamental group, and the other one on discrete\nvector fields. Some examples showing our implementation of these constructions in both\n\\sagemath\\ and \\kenzo\\ are shown, together with an approach to compute the\nhomology of the universal cover when the group is abelian even in some cases\nwhere there is no effective homology, using the twisted homology of the space.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"3 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Computing the homology of universal covers via effective homology and discrete vector fields\",\"authors\":\"Miguel Angel Marco-Buzunariz, Ana Romero\",\"doi\":\"arxiv-2409.06357\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Effective homology techniques allow us to compute homology groups of a wide\\nfamily of topological spaces. By the Whitehead tower method, this can also be\\nused to compute higher homotopy groups. However, some of these techniques (in\\nparticular, the Whitehead tower) rely on the assumption that the starting space\\nis simply connected. For some applications, this problem could be circumvented\\nby replacing the space by its universal cover, which is a simply connected\\nspace that shares the higher homotopy groups of the initial space. In this\\npaper, we formalize a simplicial construction for the universal cover, and\\nrepresent it as a twisted cartesian product. As we show with some examples, the universal cover of a space with effective\\nhomology does not necessarily have effective homology in general. We show two\\nindependent sufficient conditions that can ensure it: one is based on a\\nnilpotency property of the fundamental group, and the other one on discrete\\nvector fields. Some examples showing our implementation of these constructions in both\\n\\\\sagemath\\\\ and \\\\kenzo\\\\ are shown, together with an approach to compute the\\nhomology of the universal cover when the group is abelian even in some cases\\nwhere there is no effective homology, using the twisted homology of the space.\",\"PeriodicalId\":501119,\"journal\":{\"name\":\"arXiv - MATH - Algebraic Topology\",\"volume\":\"3 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Algebraic Topology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.06357\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Algebraic Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06357","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Computing the homology of universal covers via effective homology and discrete vector fields
Effective homology techniques allow us to compute homology groups of a wide
family of topological spaces. By the Whitehead tower method, this can also be
used to compute higher homotopy groups. However, some of these techniques (in
particular, the Whitehead tower) rely on the assumption that the starting space
is simply connected. For some applications, this problem could be circumvented
by replacing the space by its universal cover, which is a simply connected
space that shares the higher homotopy groups of the initial space. In this
paper, we formalize a simplicial construction for the universal cover, and
represent it as a twisted cartesian product. As we show with some examples, the universal cover of a space with effective
homology does not necessarily have effective homology in general. We show two
independent sufficient conditions that can ensure it: one is based on a
nilpotency property of the fundamental group, and the other one on discrete
vector fields. Some examples showing our implementation of these constructions in both
\sagemath\ and \kenzo\ are shown, together with an approach to compute the
homology of the universal cover when the group is abelian even in some cases
where there is no effective homology, using the twisted homology of the space.