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G-CW approximation
Contents
Context
Homotopy theory
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

Representation theory
representation theory

geometric representation theory

Ingredients
Definitions
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Theorems
Contents
Idea
The generalization of CW-approximation from plain homotopy theory to $G$ -equivariant homotopy theory is called $G$ -CW approximation :

For suitable equivariance groups $G$ , every topological G-space $X$ receives a $G$ -equivariant function $Q X \overset{f}{\longrightarrow} X$ from a G-CW complex $Q X$ , such that this restricts to a weak homotopy equivalence $f^H \;\colon (Q X)^H \to X^H$ on $H$ -fixed loci , for all suitable subgroups $H \subset G$ .

This should hold for $G$ a compact Lie group (such as a finite group ) and $H$ ranging over its closed subgroups .

References
Peter May et al., Thm 3.6 in: 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 )
Last revised on August 14, 2021 at 09:26:39.
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