nLab
equivariant open cover

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Representation theory

Contents

Idea

The notions of equivariant open cover and of equivariant good open cover are the generalization of the notions of open cover and good open cover from differential topology to equivariant differential topology, hence from topological spaces to topological G-spaces and G-manifolds.

Accordingly, for some equivariance group GG, an equivariant (good) open cover of a topological G-space is a (good) open cover of its underlying topological space which is compatible with the group action in a suitable way. At a minimum this means that the cover is itself a topological G-space and that the covering map is an equivariant function, but for purposes of proper equivariant homotopy theory one will typically need that passage to HH-fixed loci, for suitable subgroups HGH \subset G, preserves the (good) covering properties.

Definition

In the following, kTopSpkTopSp denotes the convenient category of compactly generated weak Hausdorff spaces.

For GGrp(kTopSp)G \,\in\, Grp(kTopSp) a (cgwh-)topological group, and HGH \,\subset\, G a subgroup, we write

(1)GAct(kTopSp)() HkTopSp G Act(kTopSp) \xrightarrow{\; (-)^H \;} kTopSp

for the functor from the category of (cgwh-)topological G-spaces which assigns HH-fixed loci.

Definition

(properly equivariant open covers)
For GGrp(SmthMfd)Grp(kTopSp)G \,\in\, Grp(SmthMfd) \xhookrightarrow{\;} Grp(kTopSp) a topological group underlying a Lie group, and GXGAct(kTopSp)G \curvearrowright X \,\in\, G Act(kTopSp) a topological G-space, say that an open cover

X^iIU ipX \widehat{X} \;\coloneqq\; \underset{i \in I}{\sqcup} U_i \;\; \xrightarrow{\;\; p \;\;} \;\; X

is

  1. equivariant (often: “invariant”) if the GG-action on XX pulls back to a GG-action on X^\widehat{X}

    GX^G(iIU i)pGX; G \curvearrowright \widehat{X} \;\coloneqq\; G \curvearrowright \big( \underset{i \in I}{\sqcup} U_i \big) \;\; \xrightarrow{\;\; p \;\;} \;\; G \curvearrowright X \,;
  2. regular if it satisfies (Yang 2014, Def. 2.7)

    (2)(a) i,jI(U i=U ji=j) (b) iIgG(U igU jU i=gU i) (c) ni 0,,i nIg 0,,g nG(U i 0U i n , g 0U i 0g nU i n gG0kngU i k=g kU i k) \begin{array}{ll} (a) & \underset{i,j \in I}{\forall} \left( U_i \,=\, U_j \;\;\;\; \Rightarrow \;\;\;\; i \,=\, j \right) \\ (b) & \underset{i \in I}{\forall} \,\, \underset{g \in G}{\forall} \; \Big( U_i \,\cap\, g \cdot U_j \;\neq\; \varnothing \;\;\Rightarrow\;\; U_i \,=\, g \cdot U_i \Big) \\ (c) & \underset{n \in \mathbb{N}}{\forall} \; \underset{ { i_0, \cdots, i_n \in I } \atop { g_0, \cdots, g_n \,\in\, G } }{\forall} \left( \begin{array}{rcl} U_{i_0} \cap \cdots \cap U_{i_n} & \neq & \varnothing \mathrlap{,} \\ g_0 \cdot U_{i_0} \cap \cdots g_n \cdot U_{i_n} & \neq & \varnothing \end{array} \;\;\Rightarrow\;\; \underset{g \in G}{\exists} \; \underset{0 \leq k \leq n}{\forall} \; g \cdot U_{i_k} \,=\, g_k \cdot U_{i_k} \right) \end{array}
  3. properly equivariant if for all compact subgroups HGH \,\subset\, G its restriction (1) to HH-fixed loci is a plain open cover of topological spaces:

    (3)HcptGX^ Hopencoverp HX H \underset{ H \underset{cpt}{\subset} G }{\forall} \;\;\; \widehat{X}^H \underoverset { open\,cover } {p^H} {\longrightarrow} X^H
  4. properly equivariant good if it is regular and proper equivariant and the restrictions (3) are all good open covers (Yang 2014, Def. 2.10).

Remark

If an equivariant open cover is regular (2) then the index set II inherits a unique GG-action igii \mapsto g \cdot i such that

gU iU g i g \cdot U_i \,\subset\, U_{g_i}

and such that the stabilizer subgroup G xGG_x \subset G of any xU iXx \in U_i \subset X also stabilizes ii:

G xU i=U i. G_x \cdot U_i \,=\, U_i \,.

Properties

Existence

Proposition

For GGrp(FinSet)Grp(Dsc)Grp(kTopSp)G \,\in\, Grp(FinSet) \xhookrightarrow{Grp(Dsc)} Grp(kTopSp) a finite group, the topological G-space underlying a smooth G-manifold admits a regular properly equivariant good open cover (Def. ).

(Yang 2014, Thm. 2.11, using the equivariant triangulation theorem)

References

Last revised on November 14, 2021 at 05:40:32. See the history of this page for a list of all contributions to it.