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representation sphere

Contents

Context

Representation theory

Spheres

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

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 December 1, 2019 at 14:15:01. See the history of this page for a list of all contributions to it.