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?
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
In the context of topology, a topological $G$-space (traditionally just $G$-space, for short, if the context is clear) is a topological space equipped with an action of a topological group $G$ (often, but crucially not always, taken to be a finite group).
The canonical homomorphisms of topological $G$-spaces are $G$-equivariant continuous functions, and the canonical choice of homotopies between these are $G$-equivariant continuous homotopies (for trivial $G$-action on the interval). A $G$-equivariant version of the Whitehead theorem says that on G-CW complexes these $G$-equivariant homotopy equivalences are equivalently those maps that induce weak homotopy equivalences on all fixed point spaces for all subgroups of $G$ (compact subgroups, if $G$ is allowed to be a Lie group).
By Elmendorf's theorem, this, in turn, is equivalent to the (∞,1)-presheaves over the orbit category of $G$. See below at In topological spaces – Homotopy theory.
See (Henriques-Gepner 07) for expression in terms of topological groupoids/orbispaces.
In the context of stable homotopy theory the stabilization of $G$-spaces is given by spectra with G-action; these lead to equivariant stable homotopy theory. See there for more details. (But beware that in this context one considers the richer concept of G-spectra, which have a forgetful functor to spectra with G-action but better homotopy theoretic properties. ) The union of this as $G$ is allowed to vary is the global equivariant stable homotopy theory.
See at equivariant Tietze extension theorem
The standard homotopy theory on $G$-spaces used in equivariant homotopy theory considers weak equivalences which are weak homotopy equivalence on all (ordinary) fixed point spaces for all suitable subgroups. By Elmendorf's theorem, this is equivalent to (∞,1)-presheaves over the orbit category of $G$.
On the other hand there is also the standard homotopy theory of infinity-actions, presented by the Borel model structure, in this context also called the “coarse” or “naive” equivariant model structure (Guillou).
We discuss some classes of examples of G-spaces.
Let $V \in RO(G)$ be an orthogonal linear representation of a finite group $G$ on a real vector space $V$. Then the underlying Euclidean space $\mathbb{R}^V$ inherits the structure of a G-space
We may call this the Euclidean G-space associated with the linear representation $V$.
Let $V \in RO(G)$ be an orthogonal linear representation of a finite group $G$ on a real vector space $V$. Then the one-point compactification of the underlying Euclidean space $\mathbb{R}^V$ inherits the structure of a G-space with the point at infinity a fixed point. This is called the $V$-representation sphere
Let $V \in RO(G)$ be an orthogonal linear representation of a finite group $G$ on a real vector space $V$.
If $G$ is the point group of a crystallographic group inside the Euclidean group
then the $G$-action on the Euclidean space $\mathbb{R}^V$ descends to the quotient by the action of the translational normal subgroup lattice $N$ (this Prop.). The resulting $G$-space is an n-torus with $G$-action, which might be called the representation torus of $V$
graphics grabbed from SS 19
Let $G$ be a finite group (or maybe a compact Lie group) and let $V$ be a $G$-linear representation over some topological ground field $k$.
Then the corresponding projective G-space is the quotient space of the complement of the origin in (the Euclidean space underlying) $V$ by the given action of the group of units of $k$ (from the $k$-vector space-structure on $V$):
and equipped with the residual $G$-action on $V$ (which passes to the quotient space since it commutes with the $k$-action, by linearity).
See at G-CW complex.
Rezk-global equivariant homotopy theory:
cohesive (∞,1)-topos | its (∞,1)-site | base (∞,1)-topos | its (∞,1)-site |
---|---|---|---|
global equivariant homotopy theory $PSh_\infty(Glo)$ | global equivariant indexing category $Glo$ | ∞Grpd $\simeq PSh_\infty(\ast)$ | point |
… sliced over terminal orbispace: $PSh_\infty(Glo)_{/\mathcal{N}}$ | $Glo_{/\mathcal{N}}$ | orbispaces $PSh_\infty(Orb)$ | global orbit category |
… sliced over $\mathbf{B}G$: $PSh_\infty(Glo)_{/\mathbf{B}G}$ | $Glo_{/\mathbf{B}G}$ | $G$-equivariant homotopy theory of G-spaces $L_{we} G Top \simeq PSh_\infty(Orb_G)$ | $G$-orbit category $Orb_{/\mathbf{B}G} = Orb_G$ |
See also
Glen Bredon, chapter II of: Introduction to compact transformation groups, Academic Press 1972
Tammo tom Dieck, Chapter 8 in: Transformation Groups and Representation Theory, Lecture Notes in Mathematics 766, Springer 1979 (doi:10.1007/BFb0085965)
Bert Guillou, A short note on models for equivariant homotopy theory (pdf)
André Henriques, David Gepner, Homotopy Theory of Orbispaces (arXiv:math/0701916)
See also the references at equivariant homotopy theory.
Last revised on November 10, 2020 at 06:03:17. See the history of this page for a list of all contributions to it.