On the homology description of equivariant unoriented bordism groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915451108327424 |
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| author | Chen, Bo Lü, Zhi |
| author_facet | Chen, Bo Lü, Zhi |
| contents | We construct a chain complex $\mathfrak{B}$ based on a double complex derived from the universal complex $X(\mathbb{Z}_2^n)$. It is shown that $\mathfrak{B}$ has a nontrivial homology only in degree $n-2$, which is isomorphic to the equivariant unoriented bordism group $\mathcal{Z}_{n+1}(\mathbb{Z}_2^n)$ of all $(n+1)$-dimensional smooth closed $\mathbb{Z}_2^n$-manifolds with isolated fixed points. By analyzing the spectral sequence of $\mathfrak{B}$, we derive a dimension formula for $\mathcal{Z}_{n+1}(\mathbb{Z}_2^n)$ as a $\mathbb{Z}_2$-vector space, which agrees with a recent result for $n=3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_11841 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the homology description of equivariant unoriented bordism groups Chen, Bo Lü, Zhi Algebraic Topology 55N22, 55M35, 57R85, 57R91, 18G40 We construct a chain complex $\mathfrak{B}$ based on a double complex derived from the universal complex $X(\mathbb{Z}_2^n)$. It is shown that $\mathfrak{B}$ has a nontrivial homology only in degree $n-2$, which is isomorphic to the equivariant unoriented bordism group $\mathcal{Z}_{n+1}(\mathbb{Z}_2^n)$ of all $(n+1)$-dimensional smooth closed $\mathbb{Z}_2^n$-manifolds with isolated fixed points. By analyzing the spectral sequence of $\mathfrak{B}$, we derive a dimension formula for $\mathcal{Z}_{n+1}(\mathbb{Z}_2^n)$ as a $\mathbb{Z}_2$-vector space, which agrees with a recent result for $n=3$. |
| title | On the homology description of equivariant unoriented bordism groups |
| topic | Algebraic Topology 55N22, 55M35, 57R85, 57R91, 18G40 |
| url | https://arxiv.org/abs/2508.11841 |