Steenrod closed $C_3$-invariant parameter ideals in the mod 2 cohomology of $\mathbb{Z}/2\times\mathbb{Z}/2$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913743738241024 |
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| author | Rüping, Henrik Stephan, Marc |
| author_facet | Rüping, Henrik Stephan, Marc |
| contents | For the nontrivial action by the cyclic group $C_3$ of order $3$ on the graded polynomial ring $\mathbb{F}_2[a,b]$, we classify the $C_3$-invariant parameter ideals that are closed under Steenrod operations. The classification has applications to free actions by the Klein four-group $\mathbb{Z}/2\times\mathbb{Z}/2$ on products of two spheres (and more generally, finite CW complexes with four-dimensional mod $2$ homology) that extend to actions by the alternating group $A_4=(\mathbb{Z}/2\times\mathbb{Z}/2)\rtimes C_3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14349 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Steenrod closed $C_3$-invariant parameter ideals in the mod 2 cohomology of $\mathbb{Z}/2\times\mathbb{Z}/2$ Rüping, Henrik Stephan, Marc Algebraic Topology Primary 55M35, Secondary 13C05, 55S10, 20J06 For the nontrivial action by the cyclic group $C_3$ of order $3$ on the graded polynomial ring $\mathbb{F}_2[a,b]$, we classify the $C_3$-invariant parameter ideals that are closed under Steenrod operations. The classification has applications to free actions by the Klein four-group $\mathbb{Z}/2\times\mathbb{Z}/2$ on products of two spheres (and more generally, finite CW complexes with four-dimensional mod $2$ homology) that extend to actions by the alternating group $A_4=(\mathbb{Z}/2\times\mathbb{Z}/2)\rtimes C_3$. |
| title | Steenrod closed $C_3$-invariant parameter ideals in the mod 2 cohomology of $\mathbb{Z}/2\times\mathbb{Z}/2$ |
| topic | Algebraic Topology Primary 55M35, Secondary 13C05, 55S10, 20J06 |
| url | https://arxiv.org/abs/2503.14349 |