$\mathbb Z_{/p}\times \mathbb Z_{/p}$ actions on $S^n\times S^n$
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866911964259680256 |
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| author | Fowler, Jim Thatcher, Courtney |
| author_facet | Fowler, Jim Thatcher, Courtney |
| contents | We determine the homotopy type of quotients of $S^n \times S^n$ by free actions of $\mathbb Z_{/p} \times \mathbb Z_{/p}$ where $2p>n+3$. Much like free $\mathbb Z_{/p}$ actions, they can be classified via the first $p$-localized $k$-invariant, but there are restrictions on the possibilities, and these restrictions are sufficient to determine every possibility in the $n=3$ case. We use this to complete the classification of free $\mathbb Z_{/p} \times \mathbb Z_{/p}$ actions on $S^3 \times S^3$, for $p>3$, by reducing the problem to the simultaneous classification of pairs of binary quadratic forms. Although the restrictions are not sufficient to determine which $k$-invariants are realizable in general, they can sometimes be used to rule out free actions by groups that contain $\mathbb Z_{/p}\times\mathbb Z_{/p}$ as a normal Abelian subgroup. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1912_12007 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | $\mathbb Z_{/p}\times \mathbb Z_{/p}$ actions on $S^n\times S^n$ Fowler, Jim Thatcher, Courtney Geometric Topology Algebraic Topology We determine the homotopy type of quotients of $S^n \times S^n$ by free actions of $\mathbb Z_{/p} \times \mathbb Z_{/p}$ where $2p>n+3$. Much like free $\mathbb Z_{/p}$ actions, they can be classified via the first $p$-localized $k$-invariant, but there are restrictions on the possibilities, and these restrictions are sufficient to determine every possibility in the $n=3$ case. We use this to complete the classification of free $\mathbb Z_{/p} \times \mathbb Z_{/p}$ actions on $S^3 \times S^3$, for $p>3$, by reducing the problem to the simultaneous classification of pairs of binary quadratic forms. Although the restrictions are not sufficient to determine which $k$-invariants are realizable in general, they can sometimes be used to rule out free actions by groups that contain $\mathbb Z_{/p}\times\mathbb Z_{/p}$ as a normal Abelian subgroup. |
| title | $\mathbb Z_{/p}\times \mathbb Z_{/p}$ actions on $S^n\times S^n$ |
| topic | Geometric Topology Algebraic Topology |
| url | https://arxiv.org/abs/1912.12007 |