The n-Category Café

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.

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?

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 :).

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.

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.

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?

4-dimensional polytopes? please explain?

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.

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.

- 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.

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(*)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.