تقرير
Equivariant dimensions of groups with operators
| العنوان: | Equivariant dimensions of groups with operators |
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| المؤلفون: | Grant, Mark, Meir, Ehud, Patchkoria, Irakli |
| سنة النشر: | 2019 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Algebraic Topology, Mathematics - Group Theory, 55N91, 20J05 (Primary), 55M30, 20E36 (Secondary) |
| الوصف: | Let $\pi$ be a group equipped with an action of a second group $G$ by automorphisms. We define the equivariant cohomological dimension ${\sf cd}_G(\pi)$, the equivariant geometric dimension ${\sf gd}_G(\pi)$, and the equivariant Lusternik-Schnirelmann category ${\sf cat}_G(\pi)$ in terms of the Bredon dimensions and classifying space of the family of subgroups of the semi-direct product $\pi\rtimes G$ consisting of sub-conjugates of $G$. When $G$ is finite, we extend theorems of Eilenberg-Ganea and Stallings-Swan to the equivariant setting, thereby showing that all three invariants coincide (except for the possibility of a $G$-group $\pi$ with ${\sf cat}_G(\pi)={\sf cd}_G(\pi)=2$ and ${\sf gd}_G(\pi)=3$). A main ingredient is the purely algebraic result that the cohomological dimension of any finite group with respect to any family of proper subgroups is greater than one. This implies a Stallings-Swan type result for families of subgroups which do not contain all finite subgroups. Comment: v3: 23 pages. Added Remark 2.7 and strengthened Example 4.3 |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/1912.01692 |
| رقم الأكسشن: | edsarx.1912.01692 |
| قاعدة البيانات: | arXiv |
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