{"title":"关于(n-2)$连接的具有无扭同调的 2n$ 维波因卡雷复合物","authors":"Xueqi Wang","doi":"arxiv-2408.09996","DOIUrl":null,"url":null,"abstract":"Let $X$ be an $(n-2)$-connected $2n$-dimensional Poincar\\'e complex with\ntorsion-free homology, where $n\\geq 4$. We prove that $X$ can be decomposed\ninto a connected sum of two Poincar\\'e complexes: one being $(n-1)$-connected,\nwhile the other having trivial $n$th homology group. Under the additional\nassumption that $H_n(X)=0$ and $Sq^2:H^{n-1}(X;\\mathbb{Z}_2)\\to\nH^{n+1}(X;\\mathbb{Z}_2)$ is trivial, we can prove that $X$ can be further\ndecomposed into connected sums of Poincar\\'e complexes whose $(n-1)$th homology\nis isomorphic to $\\mathbb{Z}$. As an application of this result, we classify\nthe homotopy types of such $2$-connected $8$-dimensional Poincar\\'e complexes.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"12 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On $(n-2)$-connected $2n$-dimensional Poincaré complexes with torsion-free homology\",\"authors\":\"Xueqi Wang\",\"doi\":\"arxiv-2408.09996\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $X$ be an $(n-2)$-connected $2n$-dimensional Poincar\\\\'e complex with\\ntorsion-free homology, where $n\\\\geq 4$. We prove that $X$ can be decomposed\\ninto a connected sum of two Poincar\\\\'e complexes: one being $(n-1)$-connected,\\nwhile the other having trivial $n$th homology group. Under the additional\\nassumption that $H_n(X)=0$ and $Sq^2:H^{n-1}(X;\\\\mathbb{Z}_2)\\\\to\\nH^{n+1}(X;\\\\mathbb{Z}_2)$ is trivial, we can prove that $X$ can be further\\ndecomposed into connected sums of Poincar\\\\'e complexes whose $(n-1)$th homology\\nis isomorphic to $\\\\mathbb{Z}$. As an application of this result, we classify\\nthe homotopy types of such $2$-connected $8$-dimensional Poincar\\\\'e complexes.\",\"PeriodicalId\":501119,\"journal\":{\"name\":\"arXiv - MATH - Algebraic Topology\",\"volume\":\"12 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-19\",\"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-2408.09996\",\"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-2408.09996","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On $(n-2)$-connected $2n$-dimensional Poincaré complexes with torsion-free homology
Let $X$ be an $(n-2)$-connected $2n$-dimensional Poincar\'e complex with
torsion-free homology, where $n\geq 4$. We prove that $X$ can be decomposed
into a connected sum of two Poincar\'e complexes: one being $(n-1)$-connected,
while the other having trivial $n$th homology group. Under the additional
assumption that $H_n(X)=0$ and $Sq^2:H^{n-1}(X;\mathbb{Z}_2)\to
H^{n+1}(X;\mathbb{Z}_2)$ is trivial, we can prove that $X$ can be further
decomposed into connected sums of Poincar\'e complexes whose $(n-1)$th homology
is isomorphic to $\mathbb{Z}$. As an application of this result, we classify
the homotopy types of such $2$-connected $8$-dimensional Poincar\'e complexes.