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nLab > Latest Changes: equivariant Hopf degree theorem

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 3rd 2019
    • (edited Mar 3rd 2019)

    am finally splitting this off from Hopf degree theorem, to make the material easier to navigate. Still much room to improve this entry further (add an actual Idea-statement to the Idea-section, add more examples, etc.)

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 3rd 2019

    What’s ℝ sqn\mathbb{R}_{sqn}? A typo, I think.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 4th 2019

    Thanks for catching that. Should have read “ℝ sgn\mathbb{R}_{sgn}” (for the sign rep). Fixed now.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2019

    In the section on the equivariant Hopf degree theorem for maps S V→S VS^V \to S^V I have expanded the previous statement

    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) }

    by making explicit that the expression in the top right is

    ∏H∈Isotr S V(G)|W G(H)|⋅π (V H)(S (V H)) \underset{ { { {H \in \mathrm{Isotr}_{S^V}(G)} } } }{\prod} \;\; {\vert W_G(H)\vert } \cdot \pi^{\left( V^H\right)} \left( S^{\left( V^H \right)} \right)

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2019
    • (edited Aug 20th 2019)

    I have added pointers to these references:

    (also at equivariant cohomotopy)

    diff, v5, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2019
    • (edited Aug 20th 2019)

    The following is a conversation I am having with David Roberts, but everyone please join in if you feel like it:

    So tomDieck 79, Sec 8.4 states the equivariant Hopf degree theorem for characterizing, say, equivariant Cohomotopy sets

    π V(X)≔π 0Maps */(X,S V) G \pi^V(X) \;\coloneqq\; \pi_0 Maps^{\ast/}(X, S^V)^G

    (for XX a pointed GG-CW-complex which matches the given representation sphere S VS^V)

    under the simplifying assumption (middle of p. 212) that the full fixed locus of XX has positive dimension: dim(X G)≥1dim\left(X^G\right) \geq 1.

    It seems pretty clear that the only difficulty for tom Dieck in removing that assumption would have been one of notation (dealing with the case distinctions that ensue), but for the record, I’d like to spell it out.

    More concretely, the issue in removing the assumption dim(X G)≥1dim\left(X^G\right) \geq 1 should all be in starting the induction (top of p. 214) in the proof of Theorem 8.4.1:

    Given a map f:X G⟶S Vf \colon X^G \longrightarrow S^V we need to see that this extends W G(H 1)W_G(H_1)-equivariantly along X 0≔X G↪X 1X_0 \coloneqq X^G \hookrightarrow X_1, where X 1⊂XX_1 \subset X is the subspace of points whose isotropy groups are conjugate to at least the second largest subgroup H 1H_1 (under some choice of linear ordering – bottom of p. 203 ).

    I think the existence of that extension is all one needs to check in order to generalize to the case that dim(X G)dim(X^G) may vanish, but let me know if I am mixed up.

    For example if X=S VX = S^V itself, the extension does exist, either as the constant map or the identity map, and we get the statement from #4.

    Now how about the case that X=(V/N) +X = ( V/N )_+ is a Euclidean GG-torus (as here)? Should be easy, but I still need to nail it down.

    And more generally?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2019

    Hm, so the plain Tietze-Gleason extension theorem does not give all possible extensions: Since the theorem needs the function (V/N) G→fS V(V/N)^G \overset{f}{\to} S^V to factor through the Euclidean GG-space V⊂S VV \subset S^V, it only serves to extend those ff which take all fixed points to 0∈V⊂S V0 \in V \subset S^V, none to ∞∈S V\infty \in S^V.

    Now, for our application it would be very interesting if it turned out that equivariant Cohomotopy classes (V/N) +⟶S V(V/N)_+ \longrightarrow S^V had to necessarily take all fixed points of (V/N)(V/N) to 0∈V0 \in V (with dim(V)≥2dim(V) \geq 2). Might this be the case?! At least, the Tietze-Gleason theorem does not say otherwise.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2019

    And while the long list of articles by Jaworowski aim to relax the assumptions of the Tietze-Gleason extension theorem, they all assume (as far as I see) that the codomain is an absolute retract (non-equivariantly), hence contractible. So these stronger extension theorems all still yield the same situation as in #7, it seems.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2019
    • (edited Aug 21st 2019)

    I see how to do it “by hand” for the special case where

    • G=ℤ 2G = \mathbb{Z}_2

    • V=ℝ sgn n≔(ℝ sgn 1) nV = \mathbb{R}^n_{\mathrm{sgn}} \coloneqq (\mathbb{R}^1_{sgn})^n a sign representation.

    The corresponding representation torus 𝕋 sgn n≔ℝ sgn n/ℤ n\mathbb{T}^n_{sgn} \coloneqq \mathbb{R}^n_{sgn}/\mathbb{Z}^n has 2 n2^n fixed points (those points all whose coordinate components are in {[0modℤ],[12modℤ]}\big\{[0 \, mod \, \mathbb{Z}],[\tfrac{1}{2}\, mod \, \mathbb{Z}] \big\}) and every small enough disk around each of these is equivariantly isomorphic to the unit disk around the origin in ℝ sgn n\mathbb{R}^n_{\mathrm{sgn}}.

    Hence at the start of the induction an equivariant map (𝕋 sgn n) +→S ℝ sgn n(\mathbb{T}^n_{sgn})_+ \to S^{\mathbb{R}^n_{sgn}} is a choice in (ℤ 2) (2 n)(\mathbb{Z}_2)^{(2^{n})} saying whether a fixed point gets send to 0 or to ∞\infty. This extends by sending a small disk around each fixed point going to 0 identically to D(ℝ sgn n)≃S sgn n∖{∞}D(\mathbb{R}^n_{sgn}) \simeq S^{n}_{\mathrm{sgn}} \setminus \{\infty\} and sending all remaining points to ∞\infty.

    An illustration is here.

    Hence, proceeding from here with tom Dieck’s proof, the full set of unstable equivariant Hopf degrees in this case should be

    π ℝ sgn n((𝕋 sgn n) +) ⟶≃deg ℤ 2 (ℤ 2) (2 n)×(2⋅ℤ) [𝕋 sgn n→cS sgn n] ↦ (deg(c ℤ 2),deg(c)) \array{ \pi^{\mathbb{R}^n_{sgn}} \big( (\mathbb{T}^n_{sgn})_+ \big) &\underoverset{\simeq}{deg_{\mathbb{Z}_2}}{\longrightarrow}& (\mathbb{Z}_2)^{(2^{n})} \times \big( 2 \cdot \mathbb{Z} \big) \\ [\mathbb{T}^n_{sgn} \overset{c}{\to} S^n_{sgn}] &\mapsto& \big( deg(c^{\mathbb{Z}_2}), deg(c) \big) }

    with the second factor ℤ\mathbb{Z} being the ordinary Hopf degree of the full underlying map.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2019

    improved illustration: here

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2019
    • (edited Oct 21st 2019)

    replaced the graphics with slightly improved ones.

    Created a new subsection for the stable version of the equivariant Hopf degree theorem, and added a graphics for this case. But no text yet.

    diff, v13, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2021

    added pointer to:

    diff, v15, current

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