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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 18th 2019

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.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 10th 2020

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 $V \oplus \mathbf{1}$:

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