Late night math thoughts April 11, 2010

David Gay

I am currently pondering the following theorem and its implications for the problem of combinatorializing/topologizing Perutz's vision of counting holomorphic multisections of broken Lefschetz fibrations: (I heard about this theorem in relation to Legendrian knots and generating families, and in particular, normal graded rulings. I also may not have this theorem exactly right, I haven't looked it up I'm just guessing at what it says.)

Theorem 1   Given an acyclic chain complex $ C_n \to \ldots \to C_1 \to C_0$ over a field, with a fixed ordered basis $ \{a_{i,1}, \ldots, a_{i,d_i}\}$ for each $ C_i$ , and with the boundary maps labelled $ \partial_i \colon\thinspace C_i \mapsto C_{i-1}$ , there is a unique lower-triangular change of basis for each $ C_i$ , giving new ordered bases $ \{a'_{i,1}, \ldots, a'_{i,d_i}\}$ , such that, for each $ (i,j)$ , either $ \partial_i (a'_{i,j}) = a'_{i-1,k}$ for some $ k$ or $ a'_{i,j} =
\partial_{i+1}(a'_{i+1,k})$ for some $ k$ . In particular, this gives a unique pairing of the bases elements.

Here, a "lower-triangular change of basis" is the kind of change of basis you have when performing handle slides in which you are not allowed to change the heights of handles: you can add multiples of a basis element $ a_{i,j}$ to a a basis element $ a_{i,k}$ only when $ k > j$ , and you cannot switch the order of basis elements. So, if this chain complex came from the gradient flow lines of a Morse function, this theorem sort of says that, over a field, and only algebraically, there is a unique way to perform handle slides without reordering the critical points, so that you end up with at most one gradient flow line between any two critical points of successive index, and that the pairing of "cancelling" critical points that results is uniquely determined by the original ordering of the corresponding critical values.

Some questions (more to follow):

If the chain complex is not acyclic, does a spin $ ^\mathbb{C}$ structure give you a way to treat it as if it were acyclic?

In the special case of a 3-manifold Morse complex in which you have removed the 0- and 3-handle, does the "admissibility" criterion in Heegard-Floer theory guarantee that every ordering of the $ \beta$ -curves gives a unique pairing between the $ \alpha$ 's and $ \beta$ 's?

David Gay 2010-04-11