It’s been a while since I posted here, partly because I was working on getting this paper ready for submission. Since I wrote about its subject in my previous post, about Derek Wise’s talk at Perimeter Institute, I’ll let that stand for now. In the meantime, we’ve had a few talks in the seminar on stacks and groupoids. Tom Prince gave a couple of interesting talks about stacks from the point of view of simplicial sheaves, explaining how they can be seen as certain categories of objects satisfying descent. Since I only have handwritten notes on this talk, and I still haven’t entirely digested it, I think I’ll talk about that at the same time as discussing the upcoming talk about descent and related stuff by José Malagon-Lopez. For right now, I’ll write about Enxin Wu’s talk on diffeological bundles and the irrational torus. (DVI notes here)  Some of the theory of diffeological spaces has been worked out by Souriau (originally) and then Patrick Iglesias-Zemmour.  Some of the categorical properties he discussed are explained by Baez and Hoffnung (Enxin’s notes give some references).  Enxin and Dan Christensen have looked a bit at diffeological spaces in the context of homotopy theory and model categories.

Part of the motivation for this seminar was to look at how groupoids and some related entities, namely stacks, and algebra in the form of noncommutative geometry (although we didn’t get as much on this as I’d hoped), can be treated as ways to expand the notion of “space”. One reason for doing this is to handle certain kinds of moduli problems, but another – more directly related to the motivation for noncommutative geometry (NCG) – is to deal with certain quotients. The irrational torus is one of these, and under the name “noncommutative torus” is a standard example in NCG. A brief introduction to it by John Baez can be found here, and more detailed discussion is in, for example, Ch3, section 2.β of Connes’ “Noncommutative Geometry“, which describes how to find its cyclic cohomology (a noncommutative anolog of cohomology of a space), which turns out to be 2-dimensional.

The point here should be to think of it as the quotient of a space by a group action. (Which gives a transformation groupoid, and from there a – noncommutative – groupoid $C^{\star}$-algebra). The space is a torus , and the group acting on it is $\mathbb{R}$ acting by translation parallel to a line with irrational slope. In particular, we can treat $T^2$ as a group $\{ (e^{ix},e^{iy}) | x,y \in \mathbb{R} \}$ with componentwise multiplication, and think of the irrational torus, given an irrational $\theta$, as the quotient $T^2/\mathbb{R}_{\theta}$ by the subgroup $\mathbb{R}_{\theta} = \{ (e^{ix},e^{i \theta x}) \}$.

Now, this is quite well-defined as a set, but as a space it’s quite horrible, even though both groups are quite nice Lie groups. In particular, the subgroup $\mathbb{R}_{\theta}$ is dense in $T^2$ – or, thought of in terms of a group acting on the torus, the orbit space of any given point is dense. So the quotient is not a manifold – in fact, it’s quite hard to visualize. This illustrates the point that smooth manifolds are badly behaved with respect to quotients. In his talk, Enxin told us about another way to approach this problem by moving to the category of diffeological spaces. As I mentioned in a previous post, this is one of a number of attempts to expand the category of smooth manifolds $\mathbf{Mfld}$, to get a category which has nice properties $\mathbf{Mfld}$ does not have, such as having quotient objects, mapping objects, and so on. Now, the category $\mathbf{Top}$ is such an example, but this loses all the information about which maps are smooth. The point is to find some intermediate generalization, which still carries information about geometry, not just topology.

A diffeological space can be defined as a concrete (i.e. $\mathbf{Set}$-valued) sheaf on the site whose objects are open neighborhoods of $\mathbb{R}^n$ (for all $n$) and whose morphisms are smooth maps, though this is sort of an abstract way to define a space. The point of it, however, is that this site gives a model of all the maps we want to call “smooth”. Defining the category $\mathbf{Diff}$ of diffeological spaces in terms of sheaves on sites helps to ensure it has nice categorical properties, but more intuitively, a smooth space $X$ is described by giving a set, and defining all the smooth maps into the space from neighborhoods of $\mathbb{R}^n$ (these are called plots, and the collection is a diffeology). This differs from a manifold, which is defined in terms of (ahem) an atlas of charts – which unlike plots are required to be local homeomorphisms into a topological space, which fit together in smooth ways. The smooth maps into $X$ also have to be compatible – which is what the condition of being a sheaf guarantees – but the point is that we no longer suppose $X$ locally looks just like $\mathbb{R}^n$, so it can include strange quotients like the irrational torus.

Now $\mathbf{Diff}$ has lots of good properties, some of which are listed in Enxin’s notes. For instance, it has all limits and colimits, and is cartesian closed. What’s more, there’s a pair of adjoint functors between $\mathbf{Top}$ and $\mathbf{Diff}$ – so there’s an “underlying topological space” for any diffeological space (a topology making all the plots continuous), and a free diffeology on any topological space (where any continuous map from a neighborhood in $\mathbb{R}^n$ is smooth). There’s also a natural diffeology on any manifold (the one generated by taking all the charts to be plots).

The real point, though, is that a lot of standard geometric constructions that are made for manifolds also make sense for diffeological spaces, so they “support geometry”. Some things which can be defined in the context of $\mathbf{Diff}$ include: dimension; tangent spaces; differential forms; cohomology; smooth homotopy groups.

Naturally, one can define a diffeological groupoid: this is just an internal groupoid in $\mathbf{Diff}$ – there are diffeological spaces $Ob$ and $Mor$ of objects and morphisms (and of course composable pairs, $Mor \times_{Ob} Mor$, which, being a limit, is also in $\mathbf{Diff}$), and the structure maps are all smooth. These are related to diffeological bundles (defined below) in that certain groupoids can be build from bundles. The resulting groupoids all have the property of being perfect, which means that $(s,t) : Mor \rightarrow Ob \times Ob$ is a subduction – i.e. is onto, and such that the product diffeology which is the natural one on $Ob \times Ob$ is also the minimal one making this map smooth.

In fact, we need this to even define diffeological bundles, which are particular kinds of surjective maps $f : X \rightarrow Y$ in $\mathbf{Diff}$. Specifically, one gets a groupoid $K_f$ whose objects are points of $Y$, and where the morphisms $hom(y,y')$ are just smooth maps from the fibre $f^{-1}(y)$ to the fibre $f^{-1}(y')$ (which, of course, are diffeological spaces because they are subsets of $X$). It’s when this groupoid is perfect that one has a bundle.

The point here is that, unlike for manifolds, we don’t have local charts, so we can’t use the definition that a bundle is “locally trivializable”, but we do have this analogous condition. In both cases, the condition implies that all the fibres are diffeomorphic to each other (in the relevant sense). Enxin also gave a few equivalent conditions, which amount to saying one gets locally trivial bundles over neighborhoods in $\mathbb{R}^n$ when pulling back $f$ along any plot.

So now we can at least point out that the irrational torus can be construed as a diffeological bundle – thinking of it as a quotient of a group by a subgroup, we can think of this as a bundle where $X = T^2$ is the total space, the base $Y$ is the space of orbits, and the fibres are all diffeomorphic to $F = \mathbb{R}_{\theta}$.

The punchline of the talk is to use this as an example which illustrates the theorem that there is a diffeological version of the long exact sequence of homotopy groups:

$\dots \rightarrow \pi_n^D(F) \rightarrow \pi_n^D(X) \rightarrow \pi_n^D(Y) \rightarrow \pi_{n-1}^D(F) \rightarrow \dots$

Using this long exact sequence, and the fact that the (diffeological) homotopy groups for manifolds (in this case, $X = T^2$ and $F = \mathbb{R}_{\theta}$ are the same as the usual ones, one can work out the homotopy groups for the base $Y$, which is the quotient $T^2/\mathbb{R}_{\theta}$. Whereas, for topological spaces, since $\mathbb{R}_{\theta}$ is dense in $T^2$, the usual homotopy groups are all zero, for diffeological spaces, we get a different answer. In particular, $\pi_1^D(Y) = \mathbb{Z} \oplus \mathbb{Z}$, a two-dimensional lattice.

It’s interesting that this essentially agrees with what noncommutative geometry tells us about the quotient, while keeping some of our plain intuitions about “space” intact – that is, without moving whole-hog into (the opposite of) a category of noncommutative algebras. It would be interesting to know how far one can push this correspondence.