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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.)
What’s ℝsqn? A typo, I think.
Thanks for catching that. Should have read “ℝsgn” (for the sign rep). Fixed now.
In the section on the equivariant Hopf degree theorem for maps SV→SV I have expanded the previous statement
πV(SV){∞}/≃⟶{ℤ2|VG=0ℤ|otherwise}×∏H∈IsotrSV(G)H≠G|WG(H)|⋅ℤ[SVc⟶SV]↦(H↦deg(cH)−offs(c,H))by making explicit that the expression in the top right is
∏H∈IsotrSV(G)|WG(H)|⋅π(VH)(S(VH))I have added pointers to these references:
Zalman Balanov, Equivariant Hopf theorem, Nonlinear Analysis: Theory, Methods & Applications Volume 30, Issue 6, December 1997, Pages 3463-3474 (doi:10.1016/S0362-546X(97)00020-5)
Davide L. Ferrario, On the equivariant Hopf theorem, Topology Volume 42, Issue 2, March 2003, Pages 447-465 (doi:10.1016/S0040-9383(02)00015-0)
(also at equivariant cohomotopy)
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
πV(X)≔π0Maps*/(X,SV)G(for X a pointed G-CW-complex which matches the given representation sphere SV)
under the simplifying assumption (middle of p. 212) that the full fixed locus of X has positive dimension: dim(XG)≥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(XG)≥1 should all be in starting the induction (top of p. 214) in the proof of Theorem 8.4.1:
Given a map f:XG⟶SV we need to see that this extends WG(H1)-equivariantly along X0≔XG↪X1, where X1⊂X is the subspace of points whose isotropy groups are conjugate to at least the second largest subgroup H1 (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(XG) may vanish, but let me know if I am mixed up.
For example if X=SV 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)+ is a Euclidean G-torus (as here)? Should be easy, but I still need to nail it down.
And more generally?
Hm, so the plain Tietze-Gleason extension theorem does not give all possible extensions: Since the theorem needs the function (V/N)Gf→SV to factor through the Euclidean G-space V⊂SV, it only serves to extend those f which take all fixed points to 0∈V⊂SV, none to ∞∈SV.
Now, for our application it would be very interesting if it turned out that equivariant Cohomotopy classes (V/N)+⟶SV had to necessarily take all fixed points of (V/N) to 0∈V (with dim(V)≥2). Might this be the case?! At least, the Tietze-Gleason theorem does not say otherwise.
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.
I see how to do it “by hand” for the special case where
G=ℤ2
V=ℝnsgn≔(ℝ1sgn)n a sign representation.
The corresponding representation torus 𝕋nsgn≔ℝnsgn/ℤn has 2n fixed points (those points all whose coordinate components are in {[0modℤ],[12modℤ]}) and every small enough disk around each of these is equivariantly isomorphic to the unit disk around the origin in ℝnsgn.
Hence at the start of the induction an equivariant map (𝕋nsgn)+→Sℝnsgn is a choice in (ℤ2)(2n) saying whether a fixed point gets send to 0 or to ∞. This extends by sending a small disk around each fixed point going to 0 identically to D(ℝnsgn)≃Snsgn∖{∞} and sending all remaining points to ∞.
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
πℝnsgn((𝕋nsgn)+)degℤ2⟶≃(ℤ2)(2n)×(2⋅ℤ)[𝕋nsgnc→Snsgn]↦(deg(cℤ2),deg(c))with the second factor ℤ being the ordinary Hopf degree of the full underlying map.
improved illustration: here
added pointer to:
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