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
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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 concept of $G$-CW complex is to that of CW-complexes as topological G-spaces are to topological spaces: for $G$ a compact topological group, the notion of $G$-CW-complex is much like that of CW-complex, only that where in the latter case one builds a topological space from gluing of disks $D^n$ (“cells”) for a $G$-CW-complex one glues products (product topological spaces) of disks with $G$-orbits $G/H$ (coset spaces) for compact subgroups $H$.
These are cofibrant objects in the $G$-fine model structure on topological G-spaces.
(…)
For finite equivariance groups $G$, a $G$-CW-complex structure may be identified with
a plain CW-complex-structure
on which the $G$-action is cellular (sends $n$-cells onto $n$-cells, respecting their boundaries)
which on cells that are sent to themselves actually restricts to the identity (i.e. cells that are fixed under the action are actually fixed point-wise).
This special case was the original definition on Bredon 1967b, I.1.
The equivariant triangulation theorem says that if a compact Lie group $G$ acts on a compact smooth manifold $X$, then the manifold admits an equivariant triangulation. In particular:
For $G$ a compact Lie group, every closed smooth G-manifold admits the structure of a G-CW complex.
Moreover, if the manifold does have a boundary, then its G-CW complex may be chosen such that the boundary is a G-subcomplex. (Illman 83, last sentence above theorem 7.1)
In particular:
(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.
For $G$ a finite group (at least), the product of two $G$-CW-complexes in compactly generated weak Hausdorff spaces is itself a $G$-CW-complex.
Since for finite $G$, a $G$-CW complex is the same as a plain CW-complex equipped with a cellular action by $G$ (Rem. ) it is clear that for this structure to be preserved by the product operation it is sufficient that the products of underlying cells constitute a CW-complex, hence that products preserve CW-complexes in compactly generated Hausdorff spaces. This is this case by this Prop..
See at equivariant cellular approximation theorem.
See at G-CW approximation.
See at equivariant Whitehead theorem.
See at Elmendorf's theorem.
The notion of G-CW complexes is, for the case of finite groups $G$, due to
announced in
In the broader generality of general topological groups and specifically of compact Lie groups, the notion of G-CW-complexes and their equivariant Whitehead theorem is due to:
Takao Matumoto, On $G$-CW complexes and a theorem of JHC Whitehead, J. Fac. Sci. Univ. Tokyo Sect. IA 18 (1971) 363-374 [irdb:00926/0001786419, PDF]
Takao Matumoto, Equivariant K-theory and Fredholm operators, J. Fac. Sci. Tokyo 18 (1971/72), 109-112 (pdf, pdf)
and, independently, due to:
Sören Illman, Chapter I of: Equivariant algebraic topology, Princeton University 1972 (pdf)
Sören Illman, Section 2 of: Equivariant singular homology and cohomology for actions of compact lie groups (doi:10.1007/BFb0070055) In: H. T. Ku, L. N. Mann, J. L. Sicks, J. C. Su (eds.), Proceedings of the Second Conference on Compact Transformation Groups Lecture Notes in Mathematics, vol 298. Springer 1972 (doi:10.1007/BFb0070029)
Sören Illman, Section 2 of: Equivariant algebraic topology, Annales de l’Institut Fourier, Tome 23 (1973) no. 2, pp. 87-91 (doi:10.5802/aif.458)
(Which, in hindsight and with Elmendorf's theorem, gives a deeper justification for the parametrization over the orbit category already proposed in Bredon 67a, Bredon 67b.)
Proof that $G$-ANRs have the equivariant homotopy type of G-CW-complexes (for $G$ a compact Lie group):
Textbook accounts:
Tammo tom Dieck, Sections I.1, I.2 of: Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)
Wolfgang Lück, Sections I.1, I.2 of: Transformation Groups and Algebraic K-Theory, Lecture Notes in Mathematics 1408 (Springer 1989) (doi:10.1007/BFb0083681)
Peter May et al., Section I.3 of: Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996 (ISBN: 978-0-8218-0319-6 pdf, pdf)
Lecture notes:
Last revised on July 9, 2022 at 21:29:43. See the history of this page for a list of all contributions to it.