nForum - Discussion Feed (Hopf degree theorem) 2022-08-12T07:12:53-04:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Urs comments on "Hopf degree theorem" (76588) https://nforum.ncatlab.org/discussion/9613/?Focus=76588#Comment_76588 2019-03-19T00:09:53-04:00 2022-08-12T07:12:53-04:00 Urs https://nforum.ncatlab.org/account/4/ 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, ...

• 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

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Urs comments on "Hopf degree theorem" (76258) https://nforum.ncatlab.org/discussion/9613/?Focus=76258#Comment_76258 2019-02-26T07:36:49-05:00 2022-08-12T07:12:53-04:00 Urs https://nforum.ncatlab.org/account/4/ 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”): for any GG-linear ...

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).

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Urs comments on "Hopf degree theorem" (76253) https://nforum.ncatlab.org/discussion/9613/?Focus=76253#Comment_76253 2019-02-26T05:41:37-05:00 2022-08-12T07:12:53-04:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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)

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Urs comments on "Hopf degree theorem" (76251) https://nforum.ncatlab.org/discussion/9613/?Focus=76251#Comment_76251 2019-02-26T04:50:55-05:00 2022-08-12T07:12:53-04:00 Urs https://nforum.ncatlab.org/account/4/ 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)&minus;1)(dim(X^H)-1)-connected (hence ...

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)

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Urs comments on "Hopf degree theorem" (76196) https://nforum.ncatlab.org/discussion/9613/?Focus=76196#Comment_76196 2019-02-20T07:54:51-05:00 2022-08-12T07:12:53-04:00 Urs https://nforum.ncatlab.org/account/4/ have generalized the statement for &pi; 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 ...

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):

$\array{ \pi^V\left( S^V\right)^{\{\infty\}/} & \overset{\simeq}{\longrightarrow} & \left\{ \array{ \mathbb{Z}_2 &\vert& V^G = 0 \\ \mathbb{Z} &\vert& \text{otherwise} } \right\} \times \underset{ { { {H \in \mathrm{Isotr}_{S^V}(G)} \atop {H \neq G} } } }{\prod} \;\; {\vert W_G(H)\vert } \cdot \mathbb{Z} \\ \big[ S^V \overset{c}{\longrightarrow} S^V \big] &\mapsto& \Big( H \mapsto \mathrm{deg} \big( c^H \big) - \mathrm{offs}(c,H) \Big) }$ ]]>
Urs comments on "Hopf degree theorem" (76169) https://nforum.ncatlab.org/discussion/9613/?Focus=76169#Comment_76169 2019-02-18T13:23:09-05:00 2022-08-12T07:12:53-04:00 Urs https://nforum.ncatlab.org/account/4/ added some lines of proof on how that example (here) follows from the general theorem diff, v7, current

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

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Urs comments on "Hopf degree theorem" (76155) https://nforum.ncatlab.org/discussion/9613/?Focus=76155#Comment_76155 2019-02-18T03:47:01-05:00 2022-08-12T07:12:53-04:00 Urs https://nforum.ncatlab.org/account/4/ Thanks for catching this! Fixed now.

Thanks for catching this! Fixed now.

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Dmitri Pavlov comments on "Hopf degree theorem" (76152) https://nforum.ncatlab.org/discussion/9613/?Focus=76152#Comment_76152 2019-02-17T23:54:08-05:00 2022-08-12T07:12:53-04:00 Dmitri Pavlov https://nforum.ncatlab.org/account/356/ In the statement of the theorem, should it really say X→X^n as opposed to X→S^n?

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

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Urs comments on "Hopf degree theorem" (76144) https://nforum.ncatlab.org/discussion/9613/?Focus=76144#Comment_76144 2019-02-17T11:39:25-05:00 2022-08-12T07:12:53-04:00 Urs https://nforum.ncatlab.org/account/4/ I have added (here) what I suppose is the resulting statement for the bipointed part of &pi; V(S V)\pi^V(S^V): &pi; V(S V) {0,&infin;}/ &longrightarrow;&simeq; ...

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

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Urs comments on "Hopf degree theorem" (76142) https://nforum.ncatlab.org/discussion/9613/?Focus=76142#Comment_76142 2019-02-17T10:02:45-05:00 2022-08-12T07:12:53-04:00 Urs https://nforum.ncatlab.org/account/4/ I’d like to remove the assumption that dim(X H)&geq;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 ...

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. (?)

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Urs comments on "Hopf degree theorem" (76137) https://nforum.ncatlab.org/discussion/9613/?Focus=76137#Comment_76137 2019-02-17T08:05:38-05:00 2022-08-12T07:12:53-04:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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…

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Urs comments on "Hopf degree theorem" (76131) https://nforum.ncatlab.org/discussion/9613/?Focus=76131#Comment_76131 2019-02-17T05:58:36-05:00 2022-08-12T07:12:53-04:00 Urs https://nforum.ncatlab.org/account/4/ giving this its own entry, for ease of referencing v1, current

giving this its own entry, for ease of referencing

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