Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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