Strongly chiral rational homology spheres with hyperbolic fundamental groups
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908488703148032 |
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| author | Neofytidis, Christoforos |
| author_facet | Neofytidis, Christoforos |
| contents | For each $m\geq0$ and any prime $p\equiv3\ \mathrm{(mod \ 4)}$, we construct strongly chiral rational homology $(4m+3)$-spheres, which have real hyperbolic fundamental groups and only non-zero integral intermediate homology groups isomorphic to $\mathbb{Z}_{2p}$ in degrees $1,2m+1$ and $4m+1$. This gives group theoretic analogues in high dimensions of the existence of strongly chiral hyperbolic rational homology $3$-spheres, as well as of the existence of strongly chiral hyperbolic manifolds of any dimension that are not rational homology spheres, which was shown by Weinberger. One of our tools will be $r$-spins. We thus investigate the relationship between the sets of degrees of self-maps of a given manifold and its $r$-spins, and give classes of manifolds for which the sets are equal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05604 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strongly chiral rational homology spheres with hyperbolic fundamental groups Neofytidis, Christoforos Geometric Topology Algebraic Topology Group Theory For each $m\geq0$ and any prime $p\equiv3\ \mathrm{(mod \ 4)}$, we construct strongly chiral rational homology $(4m+3)$-spheres, which have real hyperbolic fundamental groups and only non-zero integral intermediate homology groups isomorphic to $\mathbb{Z}_{2p}$ in degrees $1,2m+1$ and $4m+1$. This gives group theoretic analogues in high dimensions of the existence of strongly chiral hyperbolic rational homology $3$-spheres, as well as of the existence of strongly chiral hyperbolic manifolds of any dimension that are not rational homology spheres, which was shown by Weinberger. One of our tools will be $r$-spins. We thus investigate the relationship between the sets of degrees of self-maps of a given manifold and its $r$-spins, and give classes of manifolds for which the sets are equal. |
| title | Strongly chiral rational homology spheres with hyperbolic fundamental groups |
| topic | Geometric Topology Algebraic Topology Group Theory |
| url | https://arxiv.org/abs/2411.05604 |