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It’s interesting to see the back-and-forth in the definition:
Lashof 82 takes the structure group to commute with the equivariance group.
Lashof-May 86 take the structure group to extend the equivariance group.
May 90 takes the structure group to split-extend the equivariance group.
I am thinking:
The right definition of equivariant principal bundles is that which corresponds to principal bundles internal to $G$-spaces (for $G$ being the equivariance group).
With respect to this notion, it seems to me that:
Lashof 82 is too restrictive.
Lashof-May 86 is too general.
May 90 should be just right.
Does any author make this explicit: $G$-equivariant principal bundles in the semi-generalized sense used in May 90 and regarded as principal bundles internal to $G$-spaces?
Sounds like asking for the obvious – but I haven’t yet seen any article taking this point of view.
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Hm, this already has the right definition! some 15-20 years before Lashof-May. Maybe a language barrier here?
Interesting:
Now I see that Lashof 82 has a “Note added in proof” where it says that the author has meanwhile been made aware of tomDieck 69. The note goes on to credit tomDieck 69 for some theorems, but does not mention tom Dieck’s more general definition.
When Lashof-May 86 present their generalized definition, Lashof seems to have forgotten about tomDieck 69 again, because it’s not mentioned/cited.
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I have now written out some paragraphs of an Idea-section, discussing/highlighting these perspectives/issues.
In compiling this I noticed that we have relevant discussion at category of G-sets – Internal Group Actions.
I have added pointer to that from here, but I feel like giving that statement a more transparent form and home – maybe in a small entry “semidirect product groups over G are group objects in G-actions”? (small entry with a long title, that would be :-)
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(and here they finally cite tom Dieck 69! :-)
I see P Hořava in Chern-Simons gauge theory on orbifolds: Open strings from three dimensions - Journal of Geometry and Physics, 1996 opts for tom Dieck’s version:
we are summing in the “exotic” version of gauge theory over the objects classified by tom Dieck’s classifying space
referring to
Ah, that’s interesting! Thanks for the pointer. Will add it…
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Have now used the new material at equivariant group to state a definition of equivariant principal bundles simply as principal bundles internal to $G$-spaces, and to prove that this definition is equivalent to the definition in tomDieck1969.
There is room left to beautify this further. But I’ll call it quits for tonight.
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It is very interesting to read all this history of the notion of equivariant bundle in topology where the bundle is in the sense of Steenrod.
On the other hand, in algebraic geometry there is in one sense a more general notion of equivariant sheaf or a structure on a sheaf which Mumford called the G-linearization on his geometric invariant theory. The book of Mumford on GIT is from around 1965. Now it may be not general enough as it does not talk at first glance about the structure group but in fact the idea of Mumford to do the definition in the descent picture (cocycle) works at the (possibly larger) generality of fibered categories. Namely if you have a group object $G$ acting on an object $X$ in the base of a fibered category $F\to B$ where the base has fibered products, you can look at the simplicial object corresponding to the action of $G$ on $X$ (Borel construction) and look at cartesian sections of $F$ over that simplicial object. They form a category, the equivariant fiber over $X$ and its objects are $G$-equivariant objects over $X$ in the fibered category. This is, in modern language, what Mumford called the $G$-linearization in the case of the fibered category of quasicoherent sheaves (and soon used in other analogous situations). This is in a sense more general than the case of equivariant bundles.
This language has been used for example in Vistoli’s lectures on descent theory (and later in my note on some equivariant notions in noncommutative geometry in Georgian Math. J., where I point out that 2-categorical version is straightforward). Vistoli also proves the descent along torsors in this clean generality, where of course one assumes a Grothendieck topology in base and that $\pi: F\to C$ is a stack. Vistoli also shows that the equivariant object can be seen as simply an object $\xi$ in $F$ together with a natural transformation which is an action (in the category of presheaves of sets on $F$) of a group presheaf $h_G\circ \pi$ (which is obtained by taking the representing functor for $G$ on $B$ precomposed by the projection $\pi$ of the fibered category) on the representing functor $h_\xi$. This generalizes to dimension 2 by working with pseudonatural 2-transformations and actions of categorical group objects in 2-category of 2-presheaves and corresponding Yoneda.
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(only discovering now that tom Dieck has this beautiful and comprehensive textbook account of his notion of equivariant bundles)
the entry still has ellipses (…) where it refers to the local triviality condition on equivariant bundles.
All existing literature on equivariant bundles seems to require (just) that a $G$-equivariant $\mathcal{G}$-principal bundle be locally trivial as a $\mathcal{G}$-principal bundle, i.e. ignoring the action of the equivariance group $G$.
However, for a principal bundle internal to $G$-spaces the local triviality condition would be not just that the bundle is trivial on some open cover, but on some equivariant open cover (i.e. one that is compatible with the equivariance group action).
Does any author discuss this seemingly stronger condition? Is it actually equivalent to the seemingly weaker condition that everyone uses?
Oh, I see. This is the remark on p. 374 of
So he appeals to the equivariant Tietze extension theorem to find equivariant local sections around any orbit in a cover over which the bundle has already been trivialized non-equivariantly.
The equivariant extension theorem applies, when it applies, for equivariance groups being compact Lie, eg finite groups.
So Atiyah’s argument should readily generalize to all finite groups. Not so easily to compact Lie groups: While the equivariant extension theorem would still apply, we would need an argument now that the bundle still trivializes non-equivariantly over all orbits.
The other assumption hidden in Atiyah’s argument is that the equivariant extension theorem applies at all. This generally requires that our typical fiber is an absolute neighbourhood retract.
I started to make a note on examples of ANRs last night, but the my wifi collapsed. I gather examples include all finite-dimensional topological manifolds and/or all finite-dimensional locally finite CW-complexes and or all finite-dimensional (?) locally compact (?) locally contractible spaces – though the literature is incredibly shy about admitting these statements as theorems with proofs.
Thanks! I’ll be adding material to ANR now.
I feel silly for asking, but:
What’s the definition of “finite-dimensional”, used freely (without definition) by Lashof 81 (right at the beginning), also Borsuk 32 (towards the end).
This is referring to spaces that don’t seem to be assumed to be cell complexes. So what’s the tacit notion of dimension? Is it covering dimension?
Thanks for all this! I’ll try to move it all into ANR. Unless you do it first.
A useful survey
I was looking into that last night, but found it hard to spot definite results. Maybe I was too tired.
I see. Thanks!! We should record that somewhere…
No, I don’t think so.
Also, it sounds like first of all we need an entry separable metric space, so that it can host the statement of dimension of such spaces.
I see there is a comprehensive account Engelking: Dimension theory of separable metric spaces (pdf).
Thanks, very useful. So I have started separable metric space to record all this….
made explicit that principal bundles internal to $G$-spaces will have $G$-equivariant local trivialization, and added a remark (here) on sufficient conditions for plain local trivializability to imply equivariant local trivializability, following Atiyah66.
(There is room left to polish-up the remark further, but now my battery is dying…)
slightly re-worded the internal definition to cross-link explicitly with action object
On p. 28 of
appears a notion of equivariant local trivialization which only requires $G_x$-equivariant trivialization on any $G_x$-neighbourhood of $x$, for all $x \in X$.
It is not hard to show that, under standard assumptions, this is equivalent to asking for an actual $G$-equivariant local trivialization. While not hard, it’s not trivial: this needs a slice theorem, as far as I can see.
Is there any reference that would discuss this equivalence? Or just one that would state the Atiyah-Segal notion as an explicit condition one might contemplate for equivariant bundles?
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They build the existence of slices essentially into their definition of equivariant local trivialization.
Oh, I see now that the definition that Atiyah-Segal use is that given in Section 4 of Bierstone 78.
I’ll start a section on equivariant local trivializations, to sort this out…
Ah, I see, thanks.
Okay, I am on the entry now, straightening out the issue with the equivariant local trivializability.
So I guess the internalized notion of local trivialization as currently stated in the entry is actually no good, it’s too restrictive. Externally it comes out as tom Dieck’s/Bierstone’s condition but with trivial action on the direct product fiber factor.
Indeed, what actually matters for the purpose of using equivariant topological bundles as presentations for equivariant $\infty$-bundles is that the their image under passing to fixed loci and applying $Sing$ is a weakly-simplicial principal bundle (in the sense of NSS12b) in simplicial presheaves over the orbit category.
The weak-principality is automatic from the internal principality of the original bundle (that’s the point of working internally), but the remaining condition is that these bundles in simplicial presheaves be projectively Kan fibrations, and that’s the (only) aspect that any local triviality on the original topological principal bundles is needed for.
But for that we just need that all the fixed loci of the underlying topological bundle are locally trivial. And that’s what tomDieck/Bierstone’s condition ensures.
Their condition still gives a kind of internal local trivialization, but with respect to possibly more than one typical fiber (the underlying fiber is the same, but its $G$-action may change.)
Anyway, I’ll try to bring that out in the entry now.
“the following two conditions are satisfied:
(principality) the shear map”
The second condition is missing, though.
Thanks for catching this! That was a remnant of me removing the local triviality condition from the definition, yesterday.
Have just made it read “the following condition” for now. But even though it follows tradition to define equivariant principal bundles without having equivariant local triviality be part of the definition (but instead be relegated to an add-on condition later on), I find this weird and might change it back later.
I am once again inclined to think that it is tom Dieck 69 who gives the “right” definition, now of equivariant local trivialization, with all other authors ignoring that reference and incrementally rediscovering special cases of it.
But it’s a little fiddly to relate it all.
I have added a section Properties – Over coset spaces ([here](https://ncatlab.org/nlab/show/equivariant bundle#OverCosetSpaces)) and spelled out statement and proof of the characterization of equivariant principal bundles over coset spaces $G/H$ for closed subgroups.
(Following section 2.1 in tom Dieck 69, but all written out a little differently.)
I suspect now that the equivariant local triviality condition used by (at least) tom Dieck is really the following:
These are in fact internally locally trivial, but not as internal principal bunldes but as internal groupoid principal bundles, where the groupoid is that of topological groups with $G$-actions by automorphisms whose morphisms may change the automorphisms bu inner automorphisms.
Just a suspicion, still need to write out a proof.
I have added statement and proof (here) that equivariant local trivializability in the sense of Lashof implies that in the sense of tom Dieck, for $\alpha = 1$ (i.e. when the equivariance group and the structure group commute, as Lashof assumes).
(Not claiming the writeup is optimal yet. Lots of moving parts here, notation-wise. But we’ll get there.)
[Same idea should work for general $\alpha$ and comparing then to Lashof-May 86. But i’ll call it quits for tonight.]
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It’s interesting though, because if we grant that this is the origin of equivariant bundles, then the general tomDieck-definition-rediscovered-by-Lashof-May is already right there on the first page, if we agree that by “invariant subgroup” the author must mean “normal subgroup” (clearly).
Just emerged out of a little paradox crisis, with the following insight (unless I am still confused):
If we define “principal bundle” internally by just demanding the principality condition
$\Gamma \times P \underoverset{\simeq}{ (g,p) \mapsto ( p, g \cdot p ) }{\longrightarrow} P \times_X P$(which is a limit-theory condition)
and not explicitly demanding that $X \simeq P/G$ – since that is implied by the principality condition IF $P \to X$ is an effective epi — then the empty bundle is principal.
$\Gamma \times \varnothing \underoverset{\simeq}{ (g,p) \mapsto ( p, g \cdot p ) }{\longrightarrow} \varnothing \times_X \varnothing$!!
This is actually relevant – and resolves an apparent paradox – when thinking about fixed loci of equivariant bundles:
The fixed locus functor is right adjoint and hence preserves internal limit theories such as the above flavour of internal principal bundle. But it also frequently keeps the structure group intact while producing an empty underlying bundle.
First I thought I had discovered a flaw in mathematics and was about to call the Fields Institute (or who you’re gonna call in that case?) but now I see that all is good: The empty bundle is principal.
Hmm, interesting! I guess this is the difference between a torsor on the nLab (which doesn’t seem to need to be inhabited), and the more usual definition (which requires the underlying set to be inhabited).
And, luckily, the empty bundle is also a fibration…
I am puzzled by a statement on universal equivariant principal bundles. Maybe somebody can help me:
A neat explicit construction of the universal equivariant principal bundle is given in Murayama-Shimakawa 95: if the equivariance group is discrete, then (using their remark on the bottom of p. 6) the base of the universal $G$-equivariant $\Gamma$-principal bundle is the realization of the $G$-topological groupoid
$\mathcal{B} \Gamma \;\coloneqq\; TopGroupoids \big( G \times G \rightrightarrows G,\; \Gamma \rightrightarrows \ast \big)$whose $G$-action on functors $F$ and natural transformations $\eta$ is
$(g \cdot F)(g_1, g_2) \;\coloneqq\; F(g_1 g, g_2 g) \,, \;\;\;\;\;\; (g \cdot \eta)(g_1) \;\coloneqq\; \eta( g_1 g ) \,.$It’s a fun fact (which these authors don’t mention, but which one can check) that for $H \subset G$ any subgroup, the $H$-fixed groupoid of this is equivalent, as a topological groupoid (no stackification anywhere), to
$(\mathcal{B} \Gamma)^H \;\;\simeq\;\; TopGroupoids \big( G \rightrightarrows \ast,\, \Gamma \rightrightarrows \ast \big) \,.$This implies at once that the $H$-fixed subspaces of the classifying space $\left\Vert \mathcal{B}\Gamma\right\Vert$ are homotopy equivalent to the disjoint union over conjugacy classes of group homomorphisms $\rho : G \to \Gamma$ of the classifying spaces of the centralizer subgroups $\Gamma^\rho$
$(\mathcal{B} \Gamma)^H \;\;\simeq\;\; \underset{ [\rho] }{\sqcup} B \Gamma^\rho$That this should be the case is Theorem 2.17 in Lashof 82, where this is derived not from inspection of a concrete model, but from more abstract criteria for universal equivariant bundles.
There is a subtlety here in that Lashof 82 considers equivariant bundles where the equivariance group $G$ commutes with the structure group $\Gamma$, while Murayama-Shimakawa 95 mean to consider the case where both jointly act as a semidirect product group.
But in the special case where they do commute, the model of Murayama-Shimakawa 95 makes nicely manifest the fixed point structure of the classifying space for equivariant principal bundles according to Theorem 2.17 in Lashof 82.
So far so good.
But implicit in Murayama-Shimakawa 95 is that a more general action of $G$ on $\Gamma$ (to a direct product group structure) does not affect the underlying $G$-space of the universal equivariant bundle which they build.
So their result says – unless I am mixed up, but it seems clear – that the above formula for the fixed point structure actually holds generally.
Now, Lashof-May 86 generalize Lashof 82 to these more general group actions. Their Theorem 10 seems to contradict this conclusion from Murayama-Shimakawa 95:
Namely their Theorem 10 says that for (in particular) semidirect product group action $G \rtimes_\alpha \Gamma$, the $H$-fixed point subspace of the classifying space has connected components not indexed by conjugacy classes of group homomorphisms $G \to \Gamma$, as above, but by conjugacy classes of lifts of $H$ to $G \rtimes_\alpha \Gamma$ (slightly paraphrasing here).
That sounds plausible, because such subgroups are exactly what labels the “local models”, namely the equivariant bundles over $G/H$. But how is this compatible with Murayama-Shimakawa 95?
Oh, I see. I was misreading the definition of the action in Murayama-Shimakawa, p. 1293. So never mind.
starting a section on universal equivariant principal bundles (here):
added the definition of the Murayama-Shiwakawa groupoid (for discrete $G$)
$\mathcal{B}\Gamma \;\coloneqq\; Groupoids(TopSpaces) \big( G \times G \underoverset{pr_2}{pr_1}{\rightrightarrows} G, \; \Gamma \rightrightarrows \ast \big)$with its $G$-action
$(g \cdot F) (g_1, g_2) \;\coloneqq\; \alpha(g) \big( F(g_1 g, g_2 g) \big) \,, {\phantom{AAA}} (g \cdot \eta) (g_1) \;\coloneqq\; \alpha(g)(\eta(g_1))$and then statement of its $H$-fixed loci (as topological groupoids)
$\big( \mathcal{B}\Gamma \big)^H \;\; \simeq \;\; \Big( Groups(TopSpaces)_{/G} \big( G, \, \Gamma \rtimes_\alpha G \big) \Big) \sslash \Gamma \,.$I have written some words indicating the proof, which is essentially an elementary inspection (though one best uses some diagrammatic notation which I haven’t tried to Instikify here, my local version uses equations between tikcz diagrams, which cannot be imported here – and tikzcd itself cannot be nested, unfortunately)
The point is that this gives right away the fixed point behaviour of the classifying space for equivariant principal bundles according to Lashof82 Thm 2.17 and Lashof&May86 Theorem 10 (if we grant that they mean “centralizer” instead of “normalizer” in the first slot!?) – IF we can assume that passage to fixed points commutes with realization.
Now Murayama-Shiwakawa use fat realization, but comment that they could use ordinary realization (which would commute so) at least if both $G$ and $\Gamma$ are compact Lie and possibly more generally, which however they leave open.
Oh, I see that Guillou, May & Merling 17, pp. 15 has analogous discussion.
Hm, but so they can’t get around assuming $\Gamma$ to be compact Lie, either? That would be too bad.
So not even a proper action of an arbitrary Lie group?
$\Gamma$ here is the structure group, not the equivariance group.
That’s why we’d rather not have much conditions on this at all, because in practice this needs to allow for choices like $PU(\mathcal{H})$.
Hm, on the other hand the construction for $\Gamma = PU(\mathcal{H})$ in
looks just like the Murayama-Shiwakawa construction (not cited as such) but with group homomorphisms $G \to \Gamma$ restricted to “stable” maps.
Oh, I think I finally see the abstract story here:
For $\Gamma \!\sslash\! G \;\in\; Groups\big( (SingularSmoothGroupoids_\infty)_{/\mathbf{B}G} \big)$, the equivariant classifying space $\mathcal{B}_G \Gamma$ should simply be taken to be (the shape of) the right derived base change of the plain classifying stack along the unit map of the orbi-singular modality $\prec$, hence:
$\mathcal{B}_G \Gamma \;\coloneqq\; \big( \eta^{\prec}_{\mathbf{B}G} \big)_{\ast} \big( \mathbf{B} (\Gamma \!\sslash\! G) \big)$It then follows by the right base change adjunction that the geometric $H$-fixed points of $\mathcal{B}_G \Gamma$ are the homotopy $H$-fixed points of $\mathbf{B} \Gamma$, which identifies the equivariant homotopy type of $\mathcal{B}_G \Gamma$ with the Murayama-Shimakawa-style equivariant classifying space, by the observation in #73 above.
More generally, it follows by the same right base change adjunction that $G$-equivariant $\Gamma$-principal bundles classified by $\mathcal{B}_G \Gamma$ on a $G$-space $X$ are equivalently $\Gamma \sslash G$-principal bundles on the corresponding orbispace, this being the stack $X \!\sslash\! G$ in the slice over $\mathbf{B}G$ – thus identifying the the traditional theory of equivariant principal bundles with the evident stacky formulation.
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