Contact Homology

GRASP usually gives me things I can post about, and today's talk was embedded contact homology.

Start with a contact 3 manifold Y. Find it's Reeb vector field R.



Look for where this vector field makes circular orbits. Those are Reeb orbits.

Take the module generated by all unions of them.

An arbitrary element will look like

 

where R lists all the orbits and m's are the multiplicities in  .

You say the boundary of a particular union of Reeb circles is the sum over all output union of Reeb circles such that each one is counted by the size of the moduli space of holomorphic curves connecting the first set of inputs to the second set of outputs in the symplectic manifold associated to Y.

The picture looks like

The top and bottom pieces are such holomorphic curves. This therefore is showing a contribution to d^2 which we hope all cancel out.

Of course you need a condition on indices. It is so much easier to explain the index in Lagrangian Floer where it is the number of times the tangent planes turn. These indices save the day and give some control of what the end curves can be.

I don't know the index in ECH, but since he didn't say what it was in the talk, it sounds profoundly ugly. This also means I have no hope for explaining why this is a chain complex. These all have that same Morse homology flavor of trajectories breaking like shown above. In the Morse case, I can visualize all the pieces cancelling and that is where mileage can be drawn from.