added pointer to

- 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

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).

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)

]]>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)

]]>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):

$\array{ \pi^V\left( S^V\right)^{\{\infty\}/} & \overset{\simeq}{\longrightarrow} & \left\{ \array{ \mathbb{Z}_2 &\vert& V^G = 0 \\ \mathbb{Z} &\vert& \text{otherwise} } \right\} \times \underset{ { { {H \in \mathrm{Isotr}_{S^V}(G)} \atop {H \neq G} } } }{\prod} \;\; {\vert W_G(H)\vert } \cdot \mathbb{Z} \\ \big[ S^V \overset{c}{\longrightarrow} S^V \big] &\mapsto& \Big( H \mapsto \mathrm{deg} \big( c^H \big) - \mathrm{offs}(c,H) \Big) }$ ]]>added some lines of proof on how that example (here) follows from the general theorem

]]>Thanks for catching this! Fixed now.

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

]]>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

]]>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 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…

]]>giving this its own entry, for ease of referencing

]]>