nLab equivariant open cover

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 $G$, 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 $H$-fixed loci, for suitable subgroups $H \subset G$, preserves the (good) covering properties.

Definition

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

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

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

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

Definition

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

$$\widehat{X} \;\coloneqq\; \underset{i \in I}{\sqcup} U_i \;\; \xrightarrow{\;\; p \;\;} \;\; X$$

is

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

   $$G \curvearrowright \widehat{X} \;\coloneqq\; G \curvearrowright \big( \underset{i \in I}{\sqcup} U_i \big) \;\; \xrightarrow{\;\; p \;\;} \;\; G \curvearrowright X \,;$$

2. regular if it satisfies (Yang 2014, Def. 2.7)

   (2)$$\begin{array}{ll} (a) & \underset{i,j \in I}{\forall} \left( U_i \,=\, U_j \;\;\;\; \Rightarrow \;\;\;\; i \,=\, j \right) \\ (b) & \underset{i \in I}{\forall} \,\, \underset{g \in G}{\forall} \; \Big( U_i \,\cap\, g \cdot U_j \;\neq\; \varnothing \;\;\Rightarrow\;\; U_i \,=\, g \cdot U_i \Big) \\ (c) & \underset{n \in \mathbb{N}}{\forall} \; \underset{ { i_0, \cdots, i_n \in I } \atop { g_0, \cdots, g_n \,\in\, G } }{\forall} \left( \begin{array}{rcl} U_{i_0} \cap \cdots \cap U_{i_n} & \neq & \varnothing \mathrlap{,} \\ g_0 \cdot U_{i_0} \cap \cdots g_n \cdot U_{i_n} & \neq & \varnothing \end{array} \;\;\Rightarrow\;\; \underset{g \in G}{\exists} \; \underset{0 \leq k \leq n}{\forall} \; g \cdot U_{i_k} \,=\, g_k \cdot U_{i_k} \right) \end{array}$$

3. properly equivariant if for all compact subgroups $H \,\subset\, G$ its restriction (1) to $H$-fixed loci is a plain open cover of topological spaces:

   (3)$$\underset{ H \underset{cpt}{\subset} G }{\forall} \;\;\; \widehat{X}^H \underoverset { open\,cover } {p^H} {\longrightarrow} X^H$$

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 $I$ inherits a unique $G$-action $i \mapsto g \cdot i$ such that

$$g \cdot U_i \,\subset\, U_{g_i}$$

and such that the stabilizer subgroup $G_x \subset G$ of any $x \in U_i \subset X$ also stabilizes $i$:

$$G_x \cdot U_i \,=\, U_i \,.$$

Properties

Existence

Proposition

For $G \,\in\, Grp(FinSet) \xhookrightarrow{Grp(Dsc)} Grp(kTopSp)$ 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)

Related entries

• equivariant triangulation theorem

References

• Haibo Yang, Equivariant cohomology and sheaves, Journal of Algebra, 412 2014, 230-254 (doi:10.1016/j.jalgebra.2014.05.009)