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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 18th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added the statement that $G$-representation spheres are $G$-CW-complexes, with a sketch of the idea of the proof for finite groups ([here](https://ncatlab.org/nlab/show/representation+sphere#RepresentationSpheresAreGCWComplexes)) I have been looking for source (be it textbook lecture note or otherwise) that makes this statement and gives a proof in a citable way. But it seems people either like to state it as an exercise or else spell it out only in special cases. <a href="https://ncatlab.org/nlab/revision/diff/representation+sphere/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/representation+sphere/13">v13</a>, <a href="https://ncatlab.org/nlab/show/representation+sphere">current</a>

   I have added the statement that $G$-representation spheres are $G$-CW-complexes, with a sketch of the idea of the proof for finite groups (here)

   I have been looking for source (be it textbook lecture note or otherwise) that makes this statement and gives a proof in a citable way. But it seems people either like to state it as an exercise or else spell it out only in special cases.

   diff, v13, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 10th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded remark on relation to projective $G$-spaces: *** Similarly, if $V$ is 1-dimensional over the given [[ground field]] $k$, [[stereographic projection]] identifies the [[representation sphere]] of $V$ with the [[projective G-space]] of $V \oplus \mathbf{1}$: $$ \array{ V^{cpt} & \longrightarrow & k P \big( V \oplus \mathbf{1} \big) \\ v &\mapsto& \left\{ \array{ [v,1] &\vert& v \in V \\ [1,0] &\vert& v = \infty } \right. } $$ <a href="https://ncatlab.org/nlab/revision/diff/representation+sphere/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/representation+sphere/19">v19</a>, <a href="https://ncatlab.org/nlab/show/representation+sphere">current</a>

   added remark on relation to projective $G$-spaces:

   ---

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

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

   diff, v19, current