Some of my research last year had to do with notions of “positivity” in Heegaard Floer homology and knot theory (a phrase which I’m borrowing from a paper of Matt Hedden). As a simple example of positivity, a braid (and more generally, a knot) is called positive if it has a diagram containing only positive crossings:
(Positive knots become negative knots if we switch either our nomenclature or the orientation on , so there are confusions lurking everywhere in this business. Also, note that although we needed on orientation on our knot to define the sign of a crossing, the sign is actually independent of this orientation, and only depends on the embdedding of the knot in ).
A rather amazing theorem is that every positive braid is fibered. What this means is that if we take our positive braid and close it up into a knot in , then the “knot complement” , which we can think of as an open 3-manifold, actually fibers over the circle, with fiber a punctured surface. (Alternatively, if we remove a small tubular neighborhood of the knot, we can think of as a compact 3-manifold with torus boundary). Here’s how to close the braid:
In other words, there is a punctured surface , and a map , fixing the puncture, so that the knot complement is the mapping torus for , as pictured below. Such fibered 3-manifolds are very special.
Note that it is precisely because fixes the puncture that the line above closes up and becomes a knot. I should say, the proof that positive braids are fibered uses an even more amazing theorem of Stallings, which characterizes fibered knot complements in terms of a simple algebraic property of their fundamental group. This particular notion of positivity is one that appears in Matt’s paper. I recently read some work of Etienne Ghys talking about a related notion, and I thought it was so cool that I had to post about it.
Here’s the theorem, which Ghys attributes to Freed, Schwarzman, and Sullivan. Let be a compact manifold with a non-vanishing, smooth vector field . First for the background: suppose there is a closed surface which is transverse to the flow of and meets the forward orbit of every point at least once (therefore, it meets every orbit infinitely many times). We get a first return map , simply by taking and flowing it forward until it hits again, say at , and defining . Then it’s not hard to check that must be the mapping torus for , as before, and that is just the natural vector field pointing along the “time” direction of the mapping torus (up to scaling), as depicted below:
In this case, is called the suspension of . The question addressed by the theorem is: given a non-vanishing vector field on , when is it the suspension of a map ? Note that if we have such a suspension, and therefore a fibration over , we can pull back the form on to to get a closed, NON-VANISHING 1-form which is positive on . It’s not so hard to figure out that having such a form is equivalent to being a suspension. The really cool theorem is an apparently much weaker condition which is also sufficient.
The key object is the set of probability measures on which are invariant with respect to . Given any , and any 1-form , we can get a number by integrating:
This associates to any a 1-chain, i.e. something dual to a 1-form. If is exact, the above integral can be shown to be 0, using the -invariance of (use the invariance to rewrite the integrand, when is an exact form, as a total differential). Therefore, we obtain a map
whose image is in fact compact and convex. Now, if was a suspension, then actually lies entirely in some positive half-space of . Why? Well, remember that in this case we have a closed non-vanishing one-form which is positive on , and by pairing with this form we get a map which is positive on . Therefore lies in a positive half-space of . The rad theorem of FSS is that this is actually sufficient:
Theorem: is a suspension if and only if is contained in some halfspace.
One remark about the measures : note that if we have a closed periodic orbit for , i.e. some closed loop which integrates the flow, then we get a natural set of measures which are concentrated near . In this sense, should be thought of as a set of generalized periodic orbits. The measures associated to actual periodic orbits just get sent by to the class represented by these orbits in .
One way to think about the theorem, and this is how one of the proofs goes, is that just from this positive subset of homology, we can do some fancy functional analysis to create from this an actual dual form, not just a cohomology class, with the right non-vanishing and positivity properties. So somehow, we’re free to work just in homology without losing information, which seems very appealing.
I’m not yet sure exactly what this theorem useful for, but it’s so neat. In a soon to come follow up, I will talk more about positivity on the knot side, and bleg about a concrete question that I’d love to get everyone’s opinion on.