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added pointer to Bredon 72. Will add this pointer also to various related entries on equivariant homotopy theory
added pointer to:
added projective G-spaces to the list of examples
Fixed the pdf link for
(I was looking for any author who would speak about how the forgetful functor from topological $G$-spaces to topological spaces creates limits. Anyone?)
am starting a Properties-section “Change of groups and fixed loci” with the intent to spell out in full detail how any homomorphism of topological groups induces an adjoint triple of functors between the corresponding $Topological G Spaces$, and how the cross-composite of this adjoint triple with itself yields the fixed locus-adjunction.
So far I have spelled out the step that is usually being glossed over: The coinduced action adjunction (here).
(The same or analogous discussion could be at G-set or at action or even elsewhere. But for definiteness, I am now typing it out here.)
A link to the ’General abstract formulation in homotopy type theory’ section of induced representation?
Sure. Feel free to edit, I am out of the page for the moment. Typesetting the big composite adjunction locally.
Have added
(See also at induced representation for a formulation in homotopy type theory.)
I have now added the composite adjunction diagram that I was after (here):
forming fixed loci with residual Weyl-group action is the composite of a “pull-push” of (co)induced actions through the correspondence $G \leftarrow N(H) \rightarrow W(H)$ and is thus, in particular, exhibited as a right adjoint.
For what it’s worth, I have completed the discussion of the change-of-equivariance-group adjoint triple (here) with its pull-push application to Weyl-group-equivariant fixed loci (here).
Also added the pointer to Section I.1 in May 96, where this is essentially mentioned.
I tried to find it being mentioned in tom Dieck’s writings, but haven’t seen it there. Nor elsewhere, actually (e.g. not in Blumberg’s lectures).
It’s of course not a big deal. But it would seem to be the first thing you want to at least mention in any semi-comprehensive discussion of equivariant topology.
Added this statement:
Let $G$ be a compact topological group and let $f \colon X \longrightarrow Y$ be morphism of Hausdorff $G$-spaces.
Then its quotient naturality square
$\array{ X &\overset{f}{\longrightarrow}& Y \\ \big\downarrow && \big\downarrow \\ X/G &\overset{f/G}{\longrightarrow}& Y/G }$is a pullback square if and only if $f$ preserves isotropy groups.
From Prop. 4.1 in:
made more explicit in the Lemma what it means to “preserve isotropy groups”
and
added the remark that the assumption of the Lemma is met as soon as both actions are free, in which case the Lemma gives the statement familiar for morphisms between principal bundles, without however needing to assume them to be locally trivial (here)
Question:
Given a semidirect product group $\widehat G \coloneqq \Gamma \rtimes G$ and a subgroup $G_x \subset G$ with induced semidirect product subgroup $\widehat G_x \coloneqq \Gamma \rtimes G_x$, these form a commuting square of inclusions, and “induction-restriction” through this square equals “restriction-induction” through the square, in that we have a $G$-equivariant natural isomorphism of this form:
$G \times_{G_x }P \simeq \widehat G \times_{\widehat G_x} P \,.$This iso is elementary in components, but apparently it wants to lead a more high-brow life as a special case of a Beck-Chevalley property for the change-of-transformation-groups adjunction.
What’s a good general condition satisfied by this square of inclusions of subgroups under which the Beck-Chevalley property for induction/restriction of the transformations groups holds more generally? And why?
added also pointer to p. 8 of
for discussion of the change-of-groups adjoint triple
Question:
What, if any, is a large class of $G$-spaces $X$ such that their simplicial presheaf $(G/H \mapsto Sing(X^H))$ is projectively cofibrant?
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