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
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
(G-representation spheres are G-CW-complexes)
For $G$ a compact Lie group (e.g. a finite group) and $V \in RO(G)$ a finite-dimensional orthogonal $G$-linear representation, the representation sphere $S^V$ admits the structure of a G-CW-complex.
Observe that we have a $G$-equivariant homeomorphism between the representation sphere of $V$ and the unit sphere in $\mathbb{R} \oplus V$, where $\mathbb{R}$ is the 1-dimensional trivial representation (this Prop.)
It is thus sufficient to show that unit spheres in orthogonal representations admit G-CW-complex structure.
This in turn follows as soon as there is a $G$-equivariant triangulation of $S(\mathbb{R}\oplus V)$, hence a triangulation with the property that the $G$-action restricts to a bijection on its sets of $k$-dimensional cells, for each $k$. Because then if $G/H$ is an orbit of this $G$-action on the set of $k$-cells, we have a cell $G/H \times D^k$ of an induced G-CW-complex.
Since the unit spheres in (1) are smooth manifolds with smooth $G$-action, the existence of such $G$-equivariant triangulations follows for general compact Lie groups $G$ from the equivariant triangulation theorem (Illman 83).
More explicitly, in the case that $G$ is a finite group such an equivariant triangulation may be constructed as follows:
Let $\{b_1, b_2, \cdots, b_{n+1}\}$ be an orthonormal basis of $\mathbb{R} \oplus V$. Take then as vertices of the triangulation all the distinct points $\pm g(b_i) \in \mathbb{R} \oplus V$, and as edges the geodesics (great circle segments) between nearest neighbours of these points, etc.
($\mathbb{Z}_n$-CW-decomposition of 2-sphere with rotation action)
For $n \in \mathbb{N}$, $n \geq 2$, let $\mathbb{Z}_n \hookrightarrow SO(2)$ be the cyclic group acting by rotations on the plane $\mathbb{R}^2$. Writing $\mathbb{R}^2_{rot_n}$ for the corresponding representation, its representation sphere $S^{\mathbb{R}^2_{rot_n}}$ has a G-CW-complex structure as follows:
The vertices are the two fixed point poles $(G/G) \times \{0\} = \{0\}$ and $(G/G) \times \{\infty\} = \{\infty\}$;
the edges are $n$ great circle arcs obtained from any one such arc from $0$ to $\infty$ together with all its images under $G$, hence together a free $G$-orbit $(G/1) \times D^1 = G \times D^1$ of 1-cells;
the faces are the $n$ bigons between each such arc and the next one, hence together a free orbit $(G/1) \times D^2 = G \times D^2$ of 2-cells.
The graphics on the right illustrates this cell decomposition for $n = 8$:
graphics grabbed from here
Sören Illman, Smooth equivariant triangulations ofG-manifolds for $G$ a finite group, Math. Ann. (1978) 233: 199 (doi:10.1007/BF01405351)
Sören Illman, The Equivariant Triangulation Theorem for Actions of Compact Lie Groups, Mathematische Annalen (1983) Volume: 262, page 487-502 (dml:163720)
Last revised on February 26, 2019 at 10:23:44. See the history of this page for a list of all contributions to it.