geometric representation theory
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quaternionic projective line$\,\mathbb{H}P^1$
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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.
(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
is manifestly $G$-equivariant, with its inverse exhibiting $S(\mathbb{R}\oplus V)$ as the one-point compactification of $V$, hence
This also shows that $S^V$ is a smooth manifold with smooth $G$-action.
(e.g. MP 04, p. 2)
In particular:
(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 $\mathbf{1}_V \oplus \mathbf{1}$:
(e.g. Atiyah 68, Sec. 4, Greenlees 01, 9.C)
Prop. underlies the concept of equivariant complex oriented cohomology theory.
(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.
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. )
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.
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)