nLab representation sphere

Idea

Given a linear representation of a group (topological group) $G$ in a vector space/Cartesian space $\mathbb{R}^n$, then the corresponding representation sphere is the one-point compactification of this $\mathbb{R}^n$ (the $n$-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 $V$ are unit spheres in $\mathbb{R} \oplus V$)

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

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

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

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

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

This also shows that $S^V$ is a smooth manifold with smooth $G$-action.

(e.g. Marzantowicz & Prieto 2004, p. 215 (2 of 17))

Equivariant projective lines

In particular:

Proposition

(1-dimensional representation spheres are projective G-spaces)

If $\mathbf{1}_V \,\in\, G Representations_k$ is 1-dimensional over the given ground field $k$, stereographic projection identifies the representation sphere of $V$ with the projective G-space over $k$ of $\mathbf{1}_V \oplus \mathbf{1}$:

$$\array{ V^{cpt} & \longrightarrow & k P \big( \mathbf{1}_V \oplus \mathbf{1} \big) \\ v &\mapsto& \left\{ \array{ [v,1] &\vert& v \in V \\ [1,0] &\vert& v = \infty } \right. }$$

(e.g. Atiyah 68, Sec. 4, Greenlees 01, 9.C)

Prop. underlies the concept of equivariant complex oriented cohomology theory.

$G$-CW-Complex structure

Proposition

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

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

Proof

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

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

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

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

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

Let $\{b_1, b_2, \cdots, b_{n+1}\}$ be an orthonormal basis of $\mathbb{R} \oplus V$. Take then as vertices of the triangulation all the distinct points $\pm g(b_i) \in \mathbb{R} \oplus V$, 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 69 1 (2004) 214-230 [pdf, doi:10.1112/S0024610703004721]