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representation sphere

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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

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

Idea

Given a linear representation of a group (topological group) GG in a vector space/Cartesian space n\mathbb{R}^n, then the corresponding representation sphere is the one-point compactification of this n\mathbb{R}^n (the nn-sphere) regarded as a G-space.

Representation spheres induce the looping and delooping which is used in RO(G)-graded equivariant cohomology theory, represented by genuine G-spectra in equivariant stable homotopy theory.

Properties

Equivariant stereographic projection

Proposition

(representation spheres of VV are unit spheres in V\mathbb{R} \oplus V)

Let GG be a finite group and VRO(G)V \in RO(G) a finite-dimensional linear representation of GG.

Conside the unit sphere S(V)S(\mathbb{R}\oplus V) where \mathbb{R} carries the trivial representation. Then the stereographic projection homeomorphism

S(V){(1,0)}V S(\mathbb{R}\oplus V)\setminus \{(1,\mathbf{0})\} \stackrel{\simeq}{\longrightarrow} V

is manifestly GG-equivariant, with its inverse exhibiting S(V)S(\mathbb{R}\oplus V) as the one-point compactification of VV, hence

S V GS(V). S^V \simeq_G S(\mathbb{R}\oplus V) \,.

This also shows that S VS^V is a smooth manifold with smooth GG-action.

(e.g. MP 04, p. 2)

GG-CW-Complex structure

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 (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.

References

  • Andrew Blumberg, Example 1.1.5 of Equivariant homotopy theory, 2017 (pdf, GitHub)

  • Waclaw Marzantowicz, Carlos Prieto, The unstable equivariant fixed point index and the equivariant degree, Jourmal of the London Mathematical Society, Volume 69, Issue 1 February 2004 , pp. 214-230 (pdf, doi:10.1112/S0024610703004721)

Last revised on August 20, 2019 at 08:54:34. See the history of this page for a list of all contributions to it.