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I’d like to remove the assumption that , which tom Dieck makes “for simplicity” on p. 212, because then his equivariant Hopf degree theorem would at least apply to compute for non-trivial irreps .
Looking through his proof p. 213-214 it seems this should be readily accounted for simply by making the case distinction that for we start the list of degrees with a “degree” in instead of in , everything else remaining the same. (?)
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.
spelled out two examples (here and here) for pairs of -spaces and to which tom Dieck’s equivariant Hopf degree theorem applies (“matching pairs of -spaces”):
for any -linear representation
(representation sphere mapping to itself)
for the point group of a crystallographic group acting on a Euclidean space :
(torus quotient by the given lattice with its induced -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 , which tom Dieck makes “for simplicity” on p. 212, because then his equivariant Hopf degree theorem would at least apply to compute for non-trivial irreps .
Looking through his proof p. 213-214 it seems this should be readily accounted for simply by making the case distinction that for we start the list of degrees with a “degree” in instead of in , 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.
is -connected
(hence connected if , simply connected if , 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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