nLab G-representation spheres are G-CW-complexes



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(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.


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.



( 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


  • 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 10:23:44. See the history of this page for a list of all contributions to it.