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have cross-linked with Pontrjagin dual.
There’d be more to say here, have to see how much energy I’ll have.
What I am really after is whether the statement holds in crossed generality:
Are nearby crossed homomorphism, from a compact Lie group, crossed-conjugate to each other?
I hope I am understanding correctly that the codomain Lie group need not be compact.
I have added (here) the observation that nearby crossed homomorphisms out of compact Lie groups are crossed conjugate at least under the (strong) condition that the action of the domain group on the codomain group restricts to a trivial action of the center.
Not sure how useful this is, but this seems to be how far one gets by just using the statement for plain homomorphisms, i.e. without going inside its proof and trying to refine it.
I have now forwarded the question to MathOverflow:
“Are nearby crossed homomorphisms from compact Lie groups crossed-conjugate?” MO:q/403295
It seems that the proof of the plain case is tricky enough, so I guess locating some kind of internal reasoning in the slice won’t work.
What makes me expect it’s true is that the would-be implication that $CrsHom(G,\, \Gamma) = \underset{[\phi]}{\sqcup} \Gamma/Stab_{\Gamma}(\phi) \,\in\, \Gamma Act(TopSp)$ is essentially Theorem 10 in Lashof & May 1986, applied to the model of the equivariant classifying space due to Murayama & Shimakawa 1995 .
Conversely, a proof of that nearby crossed conjugacy would give a somewhat more transparent (to my mind) proof of Lashof & May’s Thm. 10 (and then also of their Thm. 11) via Murayama-Shimakawa. That’s what I am really after.
I only now fully realize that this converse argument is really the claim in the last part of Guillou, May & Merling 2017.
But it seems to me that a proof of that statement (of crossed-conjugate nearby crossed homomorphisms) is needed to complete their argument:
In sec. 4.3 p. 18 they say without further ado that the groupoid of crossed homomorphisms is “equivalent to the coproduct of its subcategories $Aut(\alpha)$” and then proceed from there. But such a decomposition does of course not exist, in general, for the topological groupoids relevant here. One needs an argument like nearby crossed homomorphisms being crossed-conjugate in order to validate this assumption.
The same kind of issue seems to affect the argument in Uribe & Lück et al, 2014:
These authors consider plain (non-crossed) homomorphism spaces, but their codomain group is $PU(\mathcal{H})$, hence not a Lie group, and hence also outside the applicability of the available proof that nearby homomorphisms are conjugate. So another argument is needed here, too.
On p. 5 these authors still highlight the topology on the space of homomorphisms, but when it comes to stating and proving (Prop. 1.6, p. 6) that the quotient by conjugation “is” a given set, it is no longer clear if the topology is taken into account, hence if discreteness of the quotient space is established (or whether that’s even considered).
[ edit: I guess here it follows from knowing (?) that the set on the right is countable? ]
But it would have to be, since later the Thm. 1.10 on p. 11 quotes the proposition in this way, and for the analogous reason as in Guillou, May & Merling, namely to argue that the fixed loci in the equivariant classifying space have connected components indexed by this set.
ah, the issue raised in #9 is addressed in the followup:
and solved on p. 38.
Am adding a remark on this to the entry now…
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