19 November 2007

Abusive dimensions and cohomology of Grassmannians

Yesterday I complained about the abuse of the word "dimension" by a lot of popular authors, namely the fact that they tried to order the dimensions.

Being a discrete mathematician, I don't think about dimensions all that often. But I'm taking a course in algebraic topology (it's required for my program); I have a habit of trying to twist the problems there into combinatorics problems whenever possible. Recently we were asked to compute the cohomology of the Grassmannian manifold GnRk. This Grassmannian is just the manifold of n-dimensional subspaces of k-dimensional space passing through the origin. It is of dimension k(n-k), as we can see by "dimension-counting". To obtain such a subspace we pick n orthogonal unit vectors in k-space. The first one is chosen from Sk-1 and thus we have k-1 "degrees of freedom", contributing k-1 to the dimension. The second one is chosen from a (k-2)-sphere in the (k-1)-dimensional orthogonal complement of this first vector; this contributes k-2 dimensions. And so on, down to k-n dimensions for the nth vector. But we've overcounted by 1 + 2 + ... + n-1, since we've actually chosen an n-dimensional flag in k dimensions; thus we need to quotient out by the orthogonal group SO(n), which has this dimension. Thus the dimension of the Grassmanian is

[(k-1) + (k-2) + ... + (k-n)] - [0 + 1 + ... + (n-1)] = n(k-n).

As you may have noticed, I used "dimension" in two senses there; the rigorous sense of the topological invariant which is always an integer, and the non-rigorous sense of "degree of freedom". At this point my complaint from yesterday is still valid, because I never ordered the dimensions, and that was the cardinal sin I attributed to people who don't know what they're talking about.

But then I caught myself saying things like "the cohomology of G2R5 in the second dimension is Z2." That's no good.

As it turns out, I rediscovered the fact that the dimension of Hm(GnRk) is the number of integer partitions of m into at most k-n parts, none of which exceed n; in the case I just mentiond we have m = 2, n = 2, k = 5, so we're looking for integer partitions of 2 into 2 parts which don't exceed 3. There are two of them, namely 2 and 1 + 1. This isn't too interesting, but say we want to know, for example, the dimension of H9(G3R10)? Then we can just write down all the partitions of 9 into three parts which are at most 7;

7+2+0, 7+1+1, 6+3+0, 6+2+1, 5+4+0, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3

and there are ten of them; thus H9(G3R10) = Z10. You can write down generating functions for these sequences of integers; I suppose one could even ask about the asymptotic behavior of these dimensions, although I don't think a topologist would care. (I can't find a source for this fact, but I'm actually pretty sure I've heard it before; the two obvious sources to me were Hatcher's Algebraic Topology and Stanley's Enumerative Combinatorics, which are both books I know reasonably well, and I found the place in each where this should be mentioned. Stanley just says that there is a connection to the cohomology of the Grassmannian and leaves it at that; Hatcher talks about symmetric polynomials but never sinks to finding actual numbers.)

But as I said, I caught myself saying things like "what's the second-dimensional cohomology of G2R5"? Second-dimensional. And I've heard this usage from Real Algebraic Topologists, too. So apparently what was bothering me yesterday was something more subtle than this abuse of language.


John Armstrong said...

I'd actually say you've got it exactly backwards: these sorts of problems in algebraic topology are what we turn combinatorics problems into. And the fact you (re)discovered is one of the nicer ones out there. It says that, in some sense, what such a partition is "really" doing is counting dimensions in the homology of a Grassmannian, and that the generating function is exactly the (graded) Euler characteristic of the homology!

To emphasize this: topologists (at least those of my stripe) indeed care very deeply about these sorts of things. Given a space you can take its sequence of homology modules. Then hang their ranks (dimensions if you're working over a field) on a power series and add them up. If you evaluate this at -1 you get the old Euler characteristic back, but if you leave it unevaluated it's sort of like taking the "cardinality" of the homology ring.

You can dig even deeper and try writing down what it would mean for a field to have one element. Then what you find is that vector spaces over this field are just pointed sets, and affine spaces are general sets, where the cardinality of the set is its dimension as a vector space. Combinatorics, in the end, is a (very weird) special case of linear algebra!

Isabel said...

What I was saying I suspected topologists didn't care about was the asymptotic behavior of those numbers. What would it even mean to say that some cohomology group of some space has "approximately 20 dimensions" because that's what some asymptotic formula says?

John Armstrong said...

If asymptotics are all you have, asymptotics are what you work with. What does it mean for a given Betti number to be "about 20"? That there are "about 20" "holes" through the space. And if you have another space that has no "holes" at all.. well then they're probably not the same space.

Isabel said...

Well, that's a good point. But do those sorts of cases where you only know the asymptotics actually occur in practice? I've never seen them, but I'm not a topologist.

Ben said...

"Second-dimensional homology" is simply wrong. I mean, I can see the logic behind it, but it's the sort of thing people should be beaten with rulers by nuns for saying. It's way too confusing. Stick to "second-degree homology" and everything will be fine.

John Armstrong said...

Ben: As opposed to "beaten by rulers with nuns", which is an accidental mental image I'm deeply in your debt for.

Ben said...

John: well...um...always happy to help?

Anonymous said...

Yes, asymptotic behaviour is important in practice. Look at the various homological stability theorems for mapping class groups used in Madsen/Weiss for good examples...