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I’d like to remove the assumption that $dim(X^H) \geq 1$, which tom Dieck makes “for simplicity” on p. 212, because then his equivariant Hopf degree theorem would at least apply to compute $\pi_V(S^V)$ for non-trivial irreps $V$.
Looking through his proof p. 213-214 it seems this should be readily accounted for simply by making the case distinction that for $H = G$ we start the list of degrees with a “degree” in $\mathbb{Z}/2$ instead of in $\mathbb{Z}$, everything else remaining the same. (?)
I have added (here) what I suppose is the resulting statement for the bipointed part of $\pi^V(S^V)$:
$\array{ \pi^V\left( S^V\right)^{\{0,\infty\}/} & \overset{\simeq}{\longrightarrow} & \underset{ { { {H \in \mathrm{Isotr}_{S^V}(G)} \atop {H \neq G} } } }{\prod} \;\; {\vert W_G(H)\vert } \cdot \mathbb{Z} \\ \big[ S^V \overset{c}{\longrightarrow} S^V \big] &\mapsto& \Big( H \mapsto \mathrm{deg} \big( c^H \big) - \mathrm{offs}(c,H) \Big) }$but check
In the statement of the theorem, should it really say X→X^n as opposed to X→S^n?
Thanks for catching this! Fixed now.
have generalized the statement for $\pi^V(S^V)$ from the bi-pointed case to the ordinary pointed case (here), which is what we really want to see, but now has this clunky case distinction in it (which is why tom Dieck’s book ignores this case):
$\array{ \pi^V\left( S^V\right)^{\{\infty\}/} & \overset{\simeq}{\longrightarrow} & \left\{ \array{ \mathbb{Z}_2 &\vert& V^G = 0 \\ \mathbb{Z} &\vert& \text{otherwise} } \right\} \times \underset{ { { {H \in \mathrm{Isotr}_{S^V}(G)} \atop {H \neq G} } } }{\prod} \;\; {\vert W_G(H)\vert } \cdot \mathbb{Z} \\ \big[ S^V \overset{c}{\longrightarrow} S^V \big] &\mapsto& \Big( H \mapsto \mathrm{deg} \big( c^H \big) - \mathrm{offs}(c,H) \Big) }$spelled out two examples (here and here) for pairs of $G$-spaces $X$ and $Y$ to which tom Dieck’s equivariant Hopf degree theorem applies (“matching pairs of $G$-spaces”):
for any $G$-linear representation
$S^V \longrightarrow S^V$
(representation sphere mapping to itself)
for $G$ the point group of a crystallographic group acting on a Euclidean space $E$:
$E/N \longrightarrow S^E$
(torus quotient by the given lattice with its induced $G$-action mapping to the representation sphere).
added pointer to
and added pointer to that textbook also at degree of a continuous function and Poincaré–Hopf theorem, and maybe elsewhere, too
I’d like to remove the assumption that $dim(X^H) \geq 1$, which tom Dieck makes “for simplicity” on p. 212, because then his equivariant Hopf degree theorem would at least apply to compute $\pi_V(S^V)$ for non-trivial irreps $V$.
Looking through his proof p. 213-214 it seems this should be readily accounted for simply by making the case distinction that for $H = G$ we start the list of degrees with a “degree” in $\mathbb{Z}/2$ instead of in $\mathbb{Z}$, everything else remaining the same. (?)
I think this was done precisely by Dieck in another reference, see 4.10 of “Transformation groups”, Walter de Gruyter \& Co., Berlin, 1987.
$Y^H$ is $(dim(X^H)-1)$-connected
(hence connected if $dim\left(X^H\right) = 1$, simply connected if $dim\left(X^H\right) = 2$, etc.);
(which is what must be meant)
This was also corrected in the mentioned reference, maybe it is worth adding it.
Thanks, that’s a valuable comment.
I might get around to looking into this later. But if you have the energy and the material all at hand, please feel invited to make an edit in the entry!
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