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G-representation spheres are G-CW-complexes

Contents

Context

Representation theory

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

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Paths and cylinders

Homotopy groups

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Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Statement

Proposition

(G-representation spheres are G-CW-complexes)

For GG a compact Lie group (e.g. a finite group) and VRO(G)V \in RO(G) a finite-dimensional orthogonal GG-linear representation, the representation sphere S VS^V admits the structure of a G-CW-complex.

Proof

Observe that we have a GG-equivariant homeomorphism between the representation sphere of VV and the unit sphere in V\mathbb{R} \oplus V, where \mathbb{R} is the 1-dimensional trivial representation (this Prop.)

(1)S VS(V). S^V \;\simeq\; S(\mathbb{R} \oplus V) \,.

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 GG-equivariant triangulation of S(V)S(\mathbb{R}\oplus V), hence a triangulation with the property that the GG-action restricts to a bijection on its sets of kk-dimensional cells, for each kk. Because then if G/HG/H is an orbit of this GG-action on the set of kk-cells, we have a cell G/H×D kG/H \times D^k of an induced G-CW-complex.

Since the unit spheres in (1) are smooth manifolds with smooth GG-action, the existence of such GG-equivariant triangulations follows for general compact Lie groups GG from the equivariant triangulation theorem (Illman 83).

More explicitly, in the case that GG is a finite group such an equivariant triangulation may be constructed as follows:

Let {b 1,b 2,,b n+1}\{b_1, b_2, \cdots, b_{n+1}\} be an orthonormal basis of V\mathbb{R} \oplus V. Take then as vertices of the triangulation all the distinct points ±g(b i)V\pm g(b_i) \in \mathbb{R} \oplus V, and as edges the geodesics (great circle segments) between nearest neighbours of these points, etc.

Examples

Example

( n\mathbb{Z}_n-CW-decomposition of 2-sphere with rotation action)

For nn \in \mathbb{N}, n2n \geq 2, let nSO(2)\mathbb{Z}_n \hookrightarrow SO(2) be the cyclic group acting by rotations on the plane 2\mathbb{R}^2. Writing rot n 2\mathbb{R}^2_{rot_n} for the corresponding representation, its representation sphere S rot n 2S^{\mathbb{R}^2_{rot_n}} has a G-CW-complex structure as follows:

  1. The vertices are the two fixed point poles (G/G)×{0}={0}(G/G) \times \{0\} = \{0\} and (G/G)×{}={}(G/G) \times \{\infty\} = \{\infty\};

  2. the edges are nn great circle arcs obtained from any one such arc from 00 to \infty together with all its images under GG, hence together a free GG-orbit (G/1)×D 1=G×D 1(G/1) \times D^1 = G \times D^1 of 1-cells;

  3. the faces are the nn bigons between each such arc and the next one, hence together a free orbit (G/1)×D 2=G×D 2(G/1) \times D^2 = G \times D^2 of 2-cells.

The graphics on the right illustrates this cell decomposition for n=8n = 8:

graphics grabbed from here

References

  • Sören Illman, Smooth equivariant triangulations ofG-manifolds for GG 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 05:23:44. See the history of this page for a list of all contributions to it.