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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 17th 2019

giving this its own entry, for ease of referencing

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeFeb 17th 2019

I have added now also the equivariant version of the Hopf degree theorem, the way tom Dieck gives it (here). It’s somewhat baroque in its list of assumption clauses. I tried to streamline for readability, but there is a limit to what one can do about it…

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeFeb 17th 2019

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. (?)

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 17th 2019

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

• CommentRowNumber5.
• CommentAuthorDmitri Pavlov
• CommentTimeFeb 18th 2019

In the statement of the theorem, should it really say X→X^n as opposed to X→S^n?

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeFeb 18th 2019
• (edited Feb 18th 2019)

Thanks for catching this! Fixed now.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeFeb 18th 2019

added some lines of proof on how that example (here) follows from the general theorem

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeFeb 20th 2019

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):

$\array{ \pi^V\left( S^V\right)^{\{\infty\}/} & \overset{\simeq}{\longrightarrow} & \left\{ \array{ \mathbb{Z}_2 &\vert& V^G = 0 \\ \mathbb{Z} &\vert& \text{otherwise} } \right\} \times \underset{ { { {H \in \mathrm{Isotr}_{S^V}(G)} \atop {H \neq G} } } }{\prod} \;\; {\vert W_G(H)\vert } \cdot \mathbb{Z} \\ \big[ S^V \overset{c}{\longrightarrow} S^V \big] &\mapsto& \Big( H \mapsto \mathrm{deg} \big( c^H \big) - \mathrm{offs}(c,H) \Big) }$
• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeFeb 26th 2019

changed the assumption statement

$Y^H$ is $dim(X^H)$-connected

(which is how tom Dieck states it here)

to

$Y^H$ 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)

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeFeb 26th 2019

I have given the list of assumptions that enter tom Dieck’s equivariant Hopf degree theorem a Definition-environment, in order to be able to better refer to it, now this Def. (“matching pair of $G$-spaces”, for lack of a better term)

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeFeb 26th 2019
• (edited Feb 26th 2019)

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”):

1. for any $G$-linear representation

$S^V \longrightarrow S^V$

(representation sphere mapping to itself)

2. 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).

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeMar 19th 2019
• (edited Mar 19th 2019)

• B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, Modern Geometry — Methods and Applications: Part II: The Geometry and Topology of Manifolds, Graduate Texts in Mathematics 104, Springer-Verlag New York, 1985

and added pointer to that textbook also at degree of a continuous function and Poincaré–Hopf theorem, and maybe elsewhere, too

• CommentRowNumber13.
• CommentAuthorstoroo
• CommentTimeSep 19th 2023

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.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeSep 19th 2023

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!