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nLab > Latest Changes: Hopf degree theorem

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 17th 2019

    giving this its own entry, for ease of referencing

    v1, current

    • 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…

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 17th 2019

    I’d like to remove the assumption that dim(X H)≥1dim(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 π V(S V)\pi_V(S^V) for non-trivial irreps VV.

    Looking through his proof p. 213-214 it seems this should be readily accounted for simply by making the case distinction that for H=GH = G we start the list of degrees with a “degree” in ℤ/2\mathbb{Z}/2 instead of in ℤ\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 π V(S V)\pi^V(S^V):

    π V(S V) {0,∞}/ ⟶≃ ∏H∈Isotr S V(G)H≠G|W G(H)|⋅ℤ [S V⟶cS V] ↦ (H↦deg(c H)−offs(c,H)) \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

    diff, v4, current

    • 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

    diff, v7, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 20th 2019

    have generalized the statement for π V(S V)\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):

    π V(S V) {∞}/ ⟶≃ {ℤ 2 | V G=0 ℤ | otherwise}×∏H∈Isotr S V(G)H≠G|W G(H)|⋅ℤ [S V⟶cS V] ↦ (H↦deg(c H)−offs(c,H)) \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) }

    diff, v14, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2019

    changed the assumption statement

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

    (which is how tom Dieck states it here)

    to

    Y HY^H is (dim(X H)−1)(dim(X^H)-1)-connected

    (hence connected if dim(X H)=1dim\left(X^H\right) = 1, simply connected if dim(X H)=2dim\left(X^H\right) = 2, etc.);

    (which is what must be meant)

    diff, v16, current

    • 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 GG-spaces”, for lack of a better term)

    diff, v17, current

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

    spelled out two examples (here and here) for pairs of GG-spaces XX and YY to which tom Dieck’s equivariant Hopf degree theorem applies (“matching pairs of GG-spaces”):

    1. for any GG-linear representation

      S V⟶S VS^V \longrightarrow S^V

      (representation sphere mapping to itself)

    2. for GG the point group of a crystallographic group acting on a Euclidean space EE:

      E/N⟶S EE/N \longrightarrow S^E

      (torus quotient by the given lattice with its induced GG-action mapping to the representation sphere).

    diff, v18, current

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

    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

    diff, v24, current

    • CommentRowNumber13.
    • CommentAuthorstoroo
    • CommentTimeSep 19th 2023

    I’d like to remove the assumption that dim(X H)≥1dim(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 π V(S V)\pi_V(S^V) for non-trivial irreps VV.

    Looking through his proof p. 213-214 it seems this should be readily accounted for simply by making the case distinction that for H=GH = G we start the list of degrees with a “degree” in ℤ/2\mathbb{Z}/2 instead of in ℤ\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 HY^H is (dim(X H)−1)(dim(X^H)-1)-connected

    (hence connected if dim(X H)=1dim\left(X^H\right) = 1, simply connected if dim(X H)=2dim\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!

1 to 14 of 14

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