A recent colloquium talk here at UWO caught my attention because it ties in quite directly to some of the things I’ve been talking about here. Alejandro Adem, from UBC (also the PIMS head-to-be) was talking about commuting n-tuples and spaces of homomorphisms. In particular, spaces of homomorphisms $HOM(\Gamma, G)$ where $\Gamma$ is a discrete group and $G$ is a Lie group. If you take $\Gamma$ to be $\mathbb{Z}^n$, then this is a space of $n$-tuples of elements of $G$ which all commute (since $\mathbb{Z}^n$ is abelian).

In particular this turns up when you want to talk about the moduli space of flat $G$-bundles on a manifold $M$, which you do in the area of TQFT’s. Flat $G$-bundles are determined by specifying holonomies in $G$ around any loop $\gamma$ – the effect of doing transport around $\gamma$. If you take the discrete group $\Gamma = \pi_1(M)$, the fundamental group of $M$, then this is an example of the kind of space Adem was talking about. In particular, speaking of commuting $n$-tuples, that $\mathbb{Z}^n$ is the even more special case when $M$ is an $n$-dimensional torus. However, it’s a tricky enough special case in its own right, as it turns out. Adem spent a fair amount of time on some of these.

In geometry, you’re perhaps more likely to be interested in the moduli space of flat bundles up to gauge equivalence – which amounts to saying that if you conjugate all your holonomies by $g$, you have an equivalent bundle. The same thing happens with spaces $HOM(\Gamma, G)$ – since $G$ acts on them by conjugation, you can take the quotient under this action. If you started with a finite group $\Gamma$, the space $HOM(\Gamma, G)$ was a manifold, but the quotient $Rep(\Gamma, G) = HOM(\Gamma,G ) / G$ may not be. However, you do have a bundle $p: HOM(\Gamma, G) \rightarrow Rep(\Gamma, G)$, so that each point in the base space is a gauge equivalence class of connections, and the fibre over each point consists of all the gauge-equivalent connections in that class.

(Throughout the talk, I found myself trying to categorify things – in building an extended TQFT, rather than a TQFT, one uses the case where \Gamma = \pi_1(M)\$). However, there you take a weak quotient, where instead of forcing gauge-equivalent objects to be equal, you just insert isomorphisms between them, getting a groupoid I’ll call $HOM(\Gamma, G) // G$. The bundle picture is related to but different from the groupoid picture. The groupoid is equivalent to its skeleton, where the objects are just the points in $Rep(\Gamma, G)$ . The morphisms at object $x$ are the group $Aut(x)$ – the points in the fibre over $x$ in the bundle $p : HOM(\Gamma, G) \rightarrow Rep(\Gamma, G)$ are all stabilized by $Aut(x)$ – it’s a coset space.

Also, when you include the morphisms, instead of looking at functions from this space into, say, $\mathbb{C}$, or $\mathbb{Z}$ – its cohomology – you tend to look at functors from the groupoid. The category of functors from it into $\mathbf{Vect}$ is exactly the 2-vector space of states it gets in the extended TQFT picture I partially described back here and here. So this is a categorified version of a cohomology module – the non-categorified version being what a regular TQFT based on gauge group $G$ would assign to $M$. I’m not sure quite how all the rest of the talk fits into this picture.)

First, though, he described some tools for dealing with such spaces. To start with, you use the classifying spaces $B\Gamma$ and $BG$ (where $BG$ is a space whose fundamental group is $G$ and which has no other interesting homotopy groups). Since “taking the classifying space” is a functor, homomorphisms $f : \Gamma \rightarrow G$ turn into continuous maps $Bf : B\Gamma \rightarrow BG$. (Even better is when $\Gamma = \pi_1(S)$ for some Riemann surface $S$ (i.e. a torus of some genus $g$), then $S$ effectively is the classifying space: $S \simeq B\pi_1(S)$). This correspondence may not be one-to-one, but the point is they tell us something about the shape of the moduli space we were interested in. Looking at homotopy classes of such $Bf$, which form a space $(B\Gamma, BG)$, we get information about the components of the moduli space – there’s a map

$E : \pi_0(HOM(\Gamma, G)) \rightarrow (B\Gamma, BG)$

which we can try to understand. Alejandro Adem then went on to use this idea to look at spaces of commuting $n$-tuples in a Lie group $G$, namely $HOM(\mathbb{Z}^n, G)$. Since the image of $\mathbb{Z}^n$ generates an Abelian subgroup of $G$, one basic result is that if every maximal such subgroup is path-connected, then so is $HOM(\mathbb{Z}^n,G)$ – there’s just one component (since any tuple can be deformed into any other). This can be extended to groups “built from” Abelian subgroups (in various ways he left undefined for this talk).

The other important tool for looking at the geometry/topology of the moduli spaces which he spoke about was (Poincaré-)Alexander-Lefschetz duality, which provides information about the topology of one space embedded in another from the topology of its complement. In particular, it gives an isomorphism between the $p^{th}$ cohomology of a space $X \subset M$ and the $(n-p)^{th}$ of its complement, where $M$ is $n$-dimensional. In particular, the spaces of commuting $n$-tuples of elements of $G$ are subspaces of the manifold $G^n$, which is much easier to understand.

So finally, among a number of other examples of how these tools come into play, the one Adem described that I was most interested in was the space $HOM(\mathbb{Z}^2,G)$, and particularly $HOM(\mathbb{Z}^2,SU(2))$, the space of $SU(2)$ connections on a torus. The complement in $SU(2)^2$ is an open set in a manifold – hence it’s a manifold itself – and in fact it turns out to be equivalent to $SU(3)$. You can get partway to seeing this by noting that the projection map $\pi_1 : SU(2)^2 \rightarrow SU(2)$ turns $SU(2)^2 - HOM(\mathbb{Z}^2,SU(2))$ into a bundle over $SU(2) - Z(SU(2))$ – the projection never hits the centre of $SU(2)$. This centre happens to be just two points, 1 and -1, leaving the base space homotopic to a sphere $S^2$. The fibre over each point $x$ is $SU(2) - Z_{SU(2)}(x)$, the whole group minus the centralizer of $x$ (i.e. everything which doesn’t commute with $x$). The centralizer of any point is just a circle, and the remaining set is homotopic to a circle itself.

So the complement of the moduli space, within $SU(2)^2$, is homotopic to a bundle of circles over a 2-sphere. There are a few of these, and it takes a little more to find out that it happens to be the 3-sphere with the Hopf fibration, but that’s what it is. Then, to find out what the moduli space itself looks like, you have to use the Alexander-Lefschetz duality. Adem didn’t show all the details, so I’m not exactly sure how, but it seems that it turns out you have a space homotopic to the one-point union of three spaces:

$SU(2) \wedge SU(2) \wedge (S^6 - SO(3))$

Now, as I said before, this is telling us information about the objects of the groupoid (also known as the moduli stack of connections), and while the morphisms shouldn’t be too hard to work out in this case, it might be nice to have a more general picture. When I raised this, Rick Jardine suggested that looking at the maps in $(B\Gamma, BG)$ should help – the classifying spaces are simplicial sets, and so is the collection of maps between them, and the above is only talking about vertex information. There should be a way of looking at $(B\Gamma, BG)$ as an infinity-category – and in this case, it should be trivial above the level of morphisms. But I don’t quite know how this works yet.