Free actions on products of real projective spaces
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912413075374080 |
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| author | Yalcin, Ergun |
| author_facet | Yalcin, Ergun |
| contents | We prove that if $G=(\mathbb{Z}/2)^r$ acts freely and cellularly on a finite-dimensional CW-complex $X$ homotopy equivalent to $\mathbb{R}P ^{n_1} \times \cdots \times \mathbb{R} P ^{n_k}$ with trivial action on the mod-$2$ cohomology, then $r \leq μ(n_1)+ \cdots + μ(n_k )$ where for each integer $n\geq 0$, $μ(n)=0$ if $n$ is even, $μ(n)=1$ if $n\equiv 1$ mod 4, and $μ(n)=2$ if $n\equiv 3$ mod 4. This proves a homotopy-theoretic version of a conjecture of Cusick. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04067 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free actions on products of real projective spaces Yalcin, Ergun Algebraic Topology Primary: 57S25, Secondary: 55T10, 20J06, 57S17 We prove that if $G=(\mathbb{Z}/2)^r$ acts freely and cellularly on a finite-dimensional CW-complex $X$ homotopy equivalent to $\mathbb{R}P ^{n_1} \times \cdots \times \mathbb{R} P ^{n_k}$ with trivial action on the mod-$2$ cohomology, then $r \leq μ(n_1)+ \cdots + μ(n_k )$ where for each integer $n\geq 0$, $μ(n)=0$ if $n$ is even, $μ(n)=1$ if $n\equiv 1$ mod 4, and $μ(n)=2$ if $n\equiv 3$ mod 4. This proves a homotopy-theoretic version of a conjecture of Cusick. |
| title | Free actions on products of real projective spaces |
| topic | Algebraic Topology Primary: 57S25, Secondary: 55T10, 20J06, 57S17 |
| url | https://arxiv.org/abs/2506.04067 |