Configuration spaces of manifolds with flows (with John Wiltshire-Gordon)

New preprint up on the arXiv:  “Algebraic structures on cohomology of configuration spaces of manifolds with flows,” a short paper joint with John Wiltshire-Gordon.

John is a student at Michigan, finishing his Ph.D. this year under David Speyer, and he’s been thinking about stuff related to FI-modules ever since his undergrad days at Chicago hanging out with Benson Farb.

But this paper isn’t actually about FI-modules!  Let me explain.  Here’s the motivating question.  When M is a manifold, and S a finite set, we denote by PConf^S M the pure configuration space of M, i.e. the space of injections from S to M.  If S is the set 1,…,n we write PConf^n M for short.

Question:  Let M be a manifold.  What natural algebraic structure is carried by the cohomology groups H^i(PConf^n M,Z)?

Here’s one structure.  If is an injection, composition yields a map from PConf^T M to PConf^S M, which i turn yields a map from H^i(PConf^S M, Z) to  H^i(PConf^T M, Z).  In other words,

is a functor from the category of finite sets with injections to the category of k-vector spaces.  Such a functor is called an FI-module over k.  A big chunk of my paper with Benson Farb and Tom Church is devoted to figuring out what consequences this structure has for the Betti numbers, and it was by these means that Tom first proved that the unordered configuration spaces have stable cohomology with rational coefficients.  (This is actually false with integral coefficients, or when the coefficient field has characteristic p, but see the beautiful theorem of Rohit Nagpal for the story about what happens in the latter case.  How have I not blogged about that already?)

So it turns out that H_i(PConf M) is a finitely generated FI-module (the definition is what you expect) and this implies that the Betti number h^i(PConf^n M) agrees with some polynomial P_i(n) for all sufficiently large n.  For example, H_1(PConf^n S^2) has dimension

(1/2)n(n-3)

for n >= 3, but not for n=0,1,2.

If you know a little more about the manifold, you can do better.  For instance, if M has a boundary component, the Betti number agrees with P_i(n) for all n.  Why?  Because there’s more algebraic structure.  You can map from PConf^T to PConf^S, above, by “forgetting” points, but you can also add points in some predetermined contractible neighborhood of the boundary.  The operation of sticking on a point * gives you a map from PConf^S to PConf^{S union *}.  (Careful, though — if you want these maps to compose nicely, you have to say all this a little more carefully, and you really only want to think of these maps as defined up to homotopy; perfectly safe as long as we’re only keeping track of the induced maps on H^i.)

We thought we had a pretty nice story:  closed manifolds have configuration spaces with eventually polynomial Betti numbers, manifolds with boundary have configuration spaces with polynomial Betti numbers on the nose.  But in practice, it seems that configuration spaces sometimes have more stability than our results guaranteed!  For instance, H_1(PConf^n S^3) has dimension

(1/2)(n-1)(n-2)

for all n>0.  And in fact EVERY Betti number of the pure configuration space of S^3 agrees with a polynomial P_i(n) for all n > 0; the results of CEF guarantee only that h^i agrees with a polynomial once n > i.

What’s going on?

In the new paper, John and I write about a different way to get “point-adding maps” on configuration space.  If your M has the good taste to have an everywhere non-vanishing vector field, you can take any one of your marked points x in M and “split it” into two points y and y’, each very near x along the flowline of the vector field, one on either side of x.  Now once again we can both add and subtract points, as in the case of open manifolds, and again this supplies the configuration spaces with a richer structure.  In fact (exercise!) H_i(PConf^n M) now carries an action of the category of noncommutative finite sets:  objects are finite sets, morphisms are set maps endowed with an ordering of each fiber.

And fortunately, John already knew a lot about the representation theory of this category and categories like it!  In particular, it follows almost immediately that, when M is a closed manifold with a vector field (like S^3) the Betti number h^i(PConf^n M) agrees with some polynomial P_i(n) for all n > 0.  (For fans of character polynomials, the character polynomial version of this holds too, for cohomology with rational coefficients.)

That’s the main idea, but there’s more stuff in the paper, including a very beautiful picture that John made which explains how to answer the question “what structure is carried by the cohomology of pure configuration space of M when M has k nonvanishing vector fields?”  The answer is FI for k=0, the category of noncommutative finite sets for k=1, and the usual category of finite sets for k > 1.