On $(n-2)$-connected $2n$-dimensional Poincaré complexes with torsion-free homology
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916361873129472 |
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| author | Wang, Xueqi |
| author_facet | Wang, Xueqi |
| contents | Let $X$ be an $(n-2)$-connected $2n$-dimensional Poincaré complex with torsion-free homology, where $n\geq 4$. We prove that $X$ can be decomposed into a connected sum of two Poincaré 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 Poincaré 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 Poincaré complexes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09996 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $(n-2)$-connected $2n$-dimensional Poincaré complexes with torsion-free homology Wang, Xueqi Algebraic Topology Let $X$ be an $(n-2)$-connected $2n$-dimensional Poincaré complex with torsion-free homology, where $n\geq 4$. We prove that $X$ can be decomposed into a connected sum of two Poincaré 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 Poincaré 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 Poincaré complexes. |
| title | On $(n-2)$-connected $2n$-dimensional Poincaré complexes with torsion-free homology |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2408.09996 |