Henry Adams, Evgeniya Lagoda, Michael Moy, Nikola Sadovek, Aditya De Saha
{"title":"持久等变同调","authors":"Henry Adams, Evgeniya Lagoda, Michael Moy, Nikola Sadovek, Aditya De Saha","doi":"arxiv-2408.17331","DOIUrl":null,"url":null,"abstract":"This article has two goals. First, we hope to give an accessible introduction\nto persistent equivariant cohomology. Given a topological group $G$ acting on a\nfiltered space, persistent Borel equivariant cohomology measures not only the\nshape of the filtration, but also attributes of the group action on the\nfiltration, including in particular its fixed points. Second, we give an\nexplicit description of the persistent equivariant cohomology of the circle\naction on the Vietoris-Rips metric thickenings of the circle, using the Serre\nspectral sequence and the Gysin homomorphism. Indeed, if $\\frac{2\\pi k}{2k+1}\n\\le r < \\frac{2\\pi(k+1)}{2k+3}$, then\n$H^*_{S^1}(\\mathrm{VR}^\\mathrm{m}(S^1;r))\\cong\n\\mathbb{Z}[u]/(1\\cdot3\\cdot5\\cdot\\ldots \\cdot (2k+1)\\, u^{k+1})$ where\n$\\mathrm{deg}(u)=2$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Persistent equivariant cohomology\",\"authors\":\"Henry Adams, Evgeniya Lagoda, Michael Moy, Nikola Sadovek, Aditya De Saha\",\"doi\":\"arxiv-2408.17331\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This article has two goals. First, we hope to give an accessible introduction\\nto persistent equivariant cohomology. Given a topological group $G$ acting on a\\nfiltered space, persistent Borel equivariant cohomology measures not only the\\nshape of the filtration, but also attributes of the group action on the\\nfiltration, including in particular its fixed points. Second, we give an\\nexplicit description of the persistent equivariant cohomology of the circle\\naction on the Vietoris-Rips metric thickenings of the circle, using the Serre\\nspectral sequence and the Gysin homomorphism. Indeed, if $\\\\frac{2\\\\pi k}{2k+1}\\n\\\\le r < \\\\frac{2\\\\pi(k+1)}{2k+3}$, then\\n$H^*_{S^1}(\\\\mathrm{VR}^\\\\mathrm{m}(S^1;r))\\\\cong\\n\\\\mathbb{Z}[u]/(1\\\\cdot3\\\\cdot5\\\\cdot\\\\ldots \\\\cdot (2k+1)\\\\, u^{k+1})$ where\\n$\\\\mathrm{deg}(u)=2$.\",\"PeriodicalId\":501444,\"journal\":{\"name\":\"arXiv - MATH - Metric Geometry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Metric Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.17331\",\"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 - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.17331","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
This article has two goals. First, we hope to give an accessible introduction
to persistent equivariant cohomology. Given a topological group $G$ acting on a
filtered space, persistent Borel equivariant cohomology measures not only the
shape of the filtration, but also attributes of the group action on the
filtration, including in particular its fixed points. Second, we give an
explicit description of the persistent equivariant cohomology of the circle
action on the Vietoris-Rips metric thickenings of the circle, using the Serre
spectral sequence and the Gysin homomorphism. Indeed, if $\frac{2\pi k}{2k+1}
\le r < \frac{2\pi(k+1)}{2k+3}$, then
$H^*_{S^1}(\mathrm{VR}^\mathrm{m}(S^1;r))\cong
\mathbb{Z}[u]/(1\cdot3\cdot5\cdot\ldots \cdot (2k+1)\, u^{k+1})$ where
$\mathrm{deg}(u)=2$.