## April 30, 2008

### Questions on 2-covers

#### Posted by David Corfield

The following may well have been talked about by John in his lectures but I didn’t see it explicitly there, so I’ll ask.

Since at level 1, we have

A Galois connection between subgroups of the fundamental group $\pi_1(X)$ and path-connected covering spaces of X for path-connected $X$. A universal covering space is simply connected.

should we not expect a level 0 analogue:

A ‘connection’ between subsets of the set of connected components $\pi_0(X)$ and covering spaces of $X$ whose locally constant fibres are either empty or $\{*\}$?

This puts these latter ‘covering spaces’ into correspondence with the poset of subsets of $\pi_0(X)$, so that the universal covering space with truth valued fibres is the empty set.

So do we have then:

A Galois 2-connection between sub-2-groups of the fundamental 2-group $\pi_2(X)$ and path-connected 2-covering spaces (with groupoid fibres) of X for path-connected $X$, a universal 2-covering space being 2-connected?

So, if we think of a nice space with nontrivial first and second homotopy, say the loop space of the 2-sphere, do we have a correspondence between 2-covering spaces and sub-2-groups of its fundamental 2-group? And is there a universal 2-connected 2-cover with 1- and 2-homotopy killed off?

Posted at April 30, 2008 8:56 AM UTC

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### Re: Questions on 2-covers

Higher Monodromy has some relevant material.

I suppose I can pose my question for $RP(2)$, with $\pi_1 = \mathbb{Z}_2$ and $\pi_2 = \mathbb{Z}$. To kill off 1-homotopy we take the universal cover $S^2$. Then I guess to kill off 2-homotopy, we head for $S^3$.

Is it then the case that there’s a correspondence between, on the one hand, 2-groups between the trivial one and the fundamental 2-group of $RP(2)$ and, on the other, 2-covering spaces between $RP(2)$ and $S^3$.

Posted by: David Corfield on April 30, 2008 2:18 PM | Permalink | Reply to this

### Re: Questions on 2-covers

I suppose it’s time to come clean - this is the content of my thesis. I came up with the universal 2-connected cover construction a year ago, and after writing the paper with Urs (which was very useful in supplying a 2-bundle with conractible total space over a 2-type), have gone back to axiomatizing 2-covering spaces and proving theorems. The fun part is making good sense of the fundamental 2-group of a topological groupoid.

This is where I feel the definition of a sub-2-group should come from, since what else are we going to use sub-2-groups for? All the Klein 2-geometry turns up as fibres of these 2-covering spaces in the guise of homogeneous groupoids (albeit more complicated than the notes appearing in another thread). I’m trying to get to Barcelona to talk about some of this at the Categorical Groups event in June, but funding falls short of the mark at the moment.

The 0-covers of a space are just (disjoint unions of) its connected components. The universal one in the pointed case is the component with the point (Whitehead’s paper “Fiber spaces and the Eilenberg Homology Groups”), and I suspect the non-pointed case is the empty space (!).

As regards your projective space example, Lens spaces would occur somewhere between the three-sphere and the two-sphere.

Posted by: David Roberts on April 30, 2008 5:41 PM | Permalink | Reply to this

### Re: Questions on 2-covers

Are we going to have to maintain radio silence then? How close are you to completion?

The $S^3$ over $RP(2)$ example looked a fun one, with pairs of interlocking circles being swept around by the fundamental 2-group of $RP(2)$. Lens spaces, eh?

I should also test out whether my guess at the Eilenberg-MacLane space that John set was right. That should have contractible 2-cover.

Posted by: David Corfield on April 30, 2008 7:19 PM | Permalink | Reply to this

### Re: Questions on 2-covers

Actually, no, I tell a lie (posting late at night is bad for your reputation, kiddies!). A Lens space is a quotient of a odd-dimensional sphere by an action of the cyclic group $\mathbf{Z}_n$. For the three-sphere and $n=2$ we get $\mathbf{R}P^3$, and there are lots of fancy generalisations. Thus what I meant to say is that Lens spaces coming from the three-sphere provide us with extra examples of this sort - they have fundamental group $\mathbf{Z}_n$ and trivial $\pi_2$, after all not so thrilling.

Posted by: David Roberts on May 1, 2008 1:35 AM | Permalink | Reply to this

### Re: Questions on 2-covers

If you take the unit sphere in $\mathbb{C}^n$ and you take a bunch of $k$th roots of unity $\omega_1, \dots, \omega_n$, the map

$(z_1, \dots, z_n) \mapsto (\omega_1 z_1, \dots, \omega_n z_n)$

generates an action of $\mathbb{Z}/k$ on this sphere, and the quotient is called a lens space. It’s a manifold since the action is free.

In the case $n = 2$, we get quotients of the 3-sphere which vary in interesting ways depending on which roots of unity we pick. In fact, we can get 3-manifolds that are homotopy equivalent but not homeomorphic!

I like the limiting case $n = \infty$, since then we get quotients of the $\infty$-sphere by actions of $\mathbb{Z}/k$. Since the $\infty$-sphere is contractible and these actions are free, any of these lens spaces will be a model of the classifying space of $\mathbb{Z}/k$. Of course the most famous example is $k = 2$. Then we get $\mathbb{RP}^\infty$.

I’m not sure why you guys are capitalizing “Lens”. These spaces weren’t invented by Professor Lens. I always thought that someone with a warped imagination imagined these spaces look a bit like a lens. Perhaps a fundamental domain for some action of $\mathbb{Z}/k$ on the 3-sphere looks a bit like a lens — it’s some sort of 3-dimensional contractible shape, isn’t it?

Posted by: John Baez on May 1, 2008 4:27 AM | Permalink | Reply to this

### Re: Questions on 2-covers

I’m not sure why you guys are capitalizing “Lens”.

For my part, I like to capitalise the first letter of a sentence.

I always thought that someone with a warped imagination imagined these spaces look a bit like a lens.

Thurston has a warped imagination?

To form a geometric model for a lens space, we need a solid something like a lens, where the angle between the upper surface and the lower surface is $2 \pi/p$…Now when the two faces are glued together, neighborhoods of $p$ points on the rim of the lens which are identified fit exactly.

This is p. 38 of Three-dimensional Geometry and Topology, where you can see a picture of the tiling of $S^3$ by lens spaces.

Perhaps you meant ‘warped’ in a geometric sense.

Posted by: David Corfield on May 1, 2008 8:52 AM | Permalink | Reply to this

### lens vs Lens

Probably something I unconsciously picked up doing my honours thesis in physics - the main paper I was working from has ‘lens’ capitalised throughout ;)

Posted by: David Roberts on May 1, 2008 3:45 PM | Permalink | Reply to this

### Re: lens vs Lens

David wrote:

For my part, I like to capitalise the first letter of a sentence.

Very clever of you to stick it there so I couldn’t tell what you were really thinking!

To form a geometric model for a lens space, we need a solid something like a lens, where the angle between the upper surface and the lower surface is $2\pi/p$… Now when the two faces are glued together, neighborhoods of $p$ points on the rim of the lens which are identified fit exactly.

Thanks, great! So, it’s not the lens space itself that looks like a lens: instead, we get a lens space by taking a convex lens and gluing its top to its bottom, rotated by a certain amount I guess.

Posted by: John Baez on May 4, 2008 6:47 AM | Permalink | Reply to this

### Re: Questions on 2-covers

$... \to Z_2 \to Z_1 \to Z_0 \to X$

says

$Z_1$ is simply-connected, and the map $Z_1 \to X$ has the homotopy properties of the universal cover of the component $Z_0$ of $X$. For larger values of $n$ one can by analogy view the map $Z_n \to X$ as an ‘$n$-connected cover’ of X. For $n \gt 1$ these do not seem to arise so frequently in nature as in the case $n = 1$. A rare exception is the Hopf map $S_3 \to S_2$

I consider this sort of use of ‘in nature’ on p. 224 of my book.

Posted by: David Corfield on May 30, 2008 4:30 PM | Permalink | Reply to this

### Re: Questions on 2-covers

This is because $Z_2 \to Z_0$ should be a 2-bundle, and we aren’t very good at finding explicit examples of 2-bundles yet :)

Pleasingly, this 2-bundle version does factor through the universal cover, so is a mini Whitehead tower consisting of just the bottom two layers (three, if $X$ isn’t path-connected). The same can be said of a general 2-covering space - it factors through a 1-covering space, analogous to the bottom layers of a Moore-Postnikov tower (no wikipedia entry for this :0 - need to get to work, people :).

Posted by: David Roberts on May 31, 2008 2:04 AM | Permalink | Reply to this

### Re: Questions on 2-covers

If you read pages 6-12 of this talk you’ll see that the ‘subgroup of $\pi_1(X)$’ way of thinking about covers quickly gets improved to the ‘transitive action of $\pi_1(X)$’ way of thinking, and ultimately the ‘functors from $\Pi_1(X)$ to $Set$’ way of thinking.

So, by the time we get around to higher categorical analogues of this, we’re not talking about stuff like sub-$2$-groups. We’re talking about stuff like functors from $\Pi_2(X)$ to $\Gpd$! And, I think this is the morally correct approach. It’s more direct and more general.

You may recall we once tried to reverse-engineer a concept of `sub-2-group’ by thinking about actions of 2-groups and their stabilizers. It was interesting and I’m sure one could go further with it. But, for classifying 2-covers, I prefer the approach in these lecture notes.

Posted by: John Baez on April 30, 2008 7:03 PM | Permalink | Reply to this

### Re: Questions on 2-covers

Paraphrasing John,

Better: ‘transitive action …’

Best: functors from $\Pi_2(X)$ to Gpd

I do agree that coverings are best classified by fibre functors, but if we can pick a few side definitions to help us categorify other things, then why not! The definition of transitive action turned out to not be what I expected, at least from what I have done with crossed modules. More playing around needs to be done.

One of my other motivations was to be able to construct explicit 2-bundles (topological or smooth) to assuage the concerns of one of my Adelaide supervisors, who would prod people about their lack of examples of nonabelian bundle gerbes.

Posted by: David Roberts on May 1, 2008 1:47 AM | Permalink | Reply to this

### Re: Questions on 2-covers

One of my other motivations was to be able to construct explicit 2-bundles (topological or smooth) to assuage the concerns of one of my Adelaide supervisors, who would prod people about their lack of examples of nonabelian bundle gerbes.

Yes, the construction and study of explicit realizations of higher nonabelian cocycles is a Good Thing.

While at the beginning there was a lack of explicit constructions, the theory of 2-bundles was always largely motivated by what turns out to be the archetypical examples: String 2-bundles.

It was pretty clear early on that these existed when the corresponding obstruction vanishes, and detailed proofs of that have recently appeared, as we know, and it was also pretty clear how to construct them explicitly as cocycles: lift a $G$-cocycle through $(U(1)\to String(G) ) \to (1 \to 1 \to G)$ which always exists, check if the obstructing $(U(1) \to 1 \to 1)$-cocycle trivializes and if so do choose a trivialization. The result is the cocycle for the String-2-bundle to be constructed.

Published writeup is lacking, of course, but there is no reason to doubt existence of explicit constructions of examples.

And this is just the first in a tower of examples: for any simply connected compact simple group $G$, we can successively ask for lifts through higher connected covers. The obstructions to these yield higher abelian cocycles canonically associated with any $G$-bundle, the vanishing of these obstruction yields explicit higher nonabelian cocycles. As mentioned in the outlook on p. 6 in our 5brane, which refers to the last part of the ndclecture.

More work on details is necessary, of course, and I am looking forward to seeing your work on this.

Posted by: Urs Schreiber on May 2, 2008 11:15 PM | Permalink | Reply to this

### Re: Questions on 2-covers

David R. wrote:

if we can pick a few side definitions to help us categorify other things, then why not!

Sure, I agree it’s good to categorify everything in the mathematical universe — all approaches to all subjects!

The definition of transitive action turned out to not be what I expected, at least from what I have done with crossed modules.

So you came up with a definition, and you checked it was right by proving some sort of theorems? Sounds interesting.

It’s too bad you may not make it to Barcelona!

Posted by: John Baez on May 1, 2008 4:39 AM | Permalink | Reply to this

### Re: Questions on 2-covers

…to construct explicit 2-bundles (topological or smooth) to assuage the concerns of one of my Adelaide supervisors…

I’ve just finished editing a first draft of a 150 minute interview I held with Barry Mazur, where ‘explicit’ features prominently. We agreed it’s meaning is quite variable, but it seems quite clear here that you supervisor wants some concrete examples.

Is there a more general thought that if higher category theory takes on the shape too much of a generalised homotopy theory, that it won’t be quite as interesting as it should?

Posted by: David Corfield on May 1, 2008 9:27 AM | Permalink | Reply to this

### Re: Questions on 2-covers

On the other hand, as David Ben-Zvi writes:

Homotopy theory is an incredibly rich beautiful subject with many overarching themes, large scale patterns, deep phenomena related to the most beautiful parts of number theory and geometry etc. It certainly has suffered from very bad salesmanship for a long time but Hopkins, Madsen, Lurie and others are changing that hopefully.

OK, perhaps a generalised homotopy theory is not such a bad thing.

Posted by: David Corfield on May 1, 2008 10:06 AM | Permalink | Reply to this

### Re: Questions on 2-covers

Looking at that first Namboodiri lecture, what would be really fun to take to the 2-level is the material on page 9, all that stuff about the symmetries of the icosahedron and $A_5$-invariant rational functions on the Riemann sphere (covered in Week 230).

Hmmm, 4-dimensional polytopes sitting in the Hopf fibration, and look at the 2-group of its symmetries, as a principal circle bundle, which fix the polytope? Then take invariant quaternion functions on $S^3$?

Posted by: David Corfield on May 1, 2008 3:03 PM | Permalink | Reply to this

### Re: Questions on 2-covers

Hmmm, 4-dimensional polytopes sitting in the Hopf fibration, and look at the 2-group of its symmetries, as a principal circle bundle, which fix the polytope?

I imagine it might be like this: Take a polytope in $\mathbf{R}^4$ whose vertices lie on the 3-sphere. I can’t recall how a principal circle bundle has a 2-group of symmetries, though. Do you mean the 2-group of equivalences of the action groupoid naturally associated to the $S^1$-space $S^3$? This has bundle automorphisms as objects - those automorphisms covering arbitrary automorphisms of the base - and $S^1$-valued functions on the total space as morphisms, satisfying a naturality condition.

Posted by: David Roberts on May 1, 2008 3:58 PM | Permalink | Reply to this

### Re: Questions on 2-covers

I’m not terribly sure this was to be taken seriously. It was just a series of analogues, but…

Those nice regular polytopes, like the 24-cell.

Yeah, then something like the 2-group you say, but keeping everything geometrically rigid.

Or could one have a polygon sitting in the circle at each fibre over a vertex of a polytope sitting in $S^2$?

It really was only a thought, but it would help to sell all this 2-group Galois business if there was some analogue of the Klein story around. But maybe that linkage of complex analysis, topology and group theory was rather special.

Posted by: David Corfield on May 1, 2008 4:45 PM | Permalink | Reply to this

### Re: Questions on 2-covers

David wrote:

It really was only a thought, but it would help to sell all this 2-group Galois business if there was some analogue of the Klein story around. But maybe that linkage of complex analysis, topology and group theory was rather special.

I like your vague idea about trying to relate 4d regular polytopes to subgroups of the unit quaternions, 2-groups and quaternionic Galois theory — though the last bit makes me want to get $\mathbb{H}P^1 = S^4$ into the game too.

I’ll think about this. Back in the lab, I’m slowly brewing a few concoctions which may or may not be related — I’ll have to test them out on you at some point.

Posted by: John Baez on June 2, 2008 8:09 AM | Permalink | Reply to this

### Re: Questions on 2-covers

Regarding $\mathbb{H} P^1$, if that were to work like the complex case, you’d think you’d still be in the realm of 1-groups. A small test of this would be to write a similar account for $\mathbb{R} P^1 \cong S^1$.

Hmm, does one ever consider the domain and codomain of rational real functions as the ‘Riemann’ circle?

Would that go through about rational real functions fixing, say, a regular pentagon in the circle?

The reason I looked to $S^3$ was to get something bundle-like onto the scene, where perhaps 2-groups would be more at home.

Surely someone has already looked at rational quaternion functions on $S^4$.

Posted by: David Corfield on June 2, 2008 11:15 AM | Permalink | Reply to this

### Re: Questions on 2-covers

I know Jim recommended we take a look at Noohi’s papers here. But maybe the recommendation should be repeated in this context, i.e., for how 2-groups mix with the analytic, etc., see Uniformization of Deligne-Mumford Stacks (with Behrend).

Along with Notes on 2-groupoids, 2-groups, and crossed-modules, Noohi also has a paper on weighted projective general linear 2-groups, (less general treatment in section 8 of ‘Uniformization’ paper). Unweighted such things were very much on the Klein 2-geometry agenda at one point.

Posted by: David Corfield on May 2, 2008 10:27 AM | Permalink | Reply to this

### Re: Questions on 2-covers

John said:

We’re talking about stuff like functors from $\Pi_2(X)$ to Gpd!

Does this stuff work equally well for any flavour of groupoid – topological, Lie, etc.?

Posted by: David Corfield on May 5, 2008 5:45 PM | Permalink | Reply to this

### Re: Questions on 2-covers

- topological, Lie, etc.?

not as easy as all that! The “right” bicategory of groupoids internal to a category without Choice is one formed by inverting the fully faithful, essentially epimorphic(*) functors. Even then, covers with fibres that are Lie groupoids are really just smooth 2-bundles. From a Pursuing Stacks POV we want 1-types as fibres, BUT we can do this in a sneaky way:

Q: What are the fibres of covering spaces?

A: Objects of the essential image of the full functor $(\mathbf{Set} \to \mathbf{Top})$

I have a question for Urs about a certain type of Lie 2-group, which I will ask by email and further resolve this issue.

Oh, and as for radio silence, I toyed with the idea of doing a Schreiber and working through my thesis online as it comes, but I don’t think I can. General discussion is of course ok, and I’ll drop hints, but when something is solidly written I’ll pass it round to interested persons.

————-

(*)A new phrase I needed to coin: essentially surjective such that the “surjectve” part is an appropriate epimorphism: local sections for $\mathbf{Top}$, submersion for Lie.

Posted by: David Roberts on May 6, 2008 2:50 AM | Permalink | Reply to this

### Re: Questions on 2-covers

I wonder if “doing a Schreiber” will make its way into the lexicon.

Of course, you’re always welcome to give us guest posts.

Posted by: David Corfield on May 7, 2008 12:48 PM | Permalink | Reply to this

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