topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
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.
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
for the functor from the category of (cgwh-)topological G-spaces which assigns $H$-fixed loci.
(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
is
equivariant (often: “invariant”) if the $G$-action on $X$ pulls back to a $G$-action on $\widehat{X}$
regular if it satisfies (Yang 2014, Def. 2.7)
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:
properly equivariant good if it is regular and proper equivariant and the restrictions (3) are all good open covers (Yang 2014, Def. 2.10).
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
and such that the stabilizer subgroup $G_x \subset G$ of any $x \in U_i \subset X$ also stabilizes $i$:
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)
Last revised on November 14, 2021 at 10:40:32. See the history of this page for a list of all contributions to it.