Is there "nonorientable Heegaard Floer homology"?

Question

I have a Heegaard diagram which produces a non-orientable 3-manifold. I want to know any 3-manifold invariant which can be calculated from Heegaard diagrams for non-orientable 3-manifold. (As far as I know, Roklin invariant or Heegaard Floer homologies are defined on oriented 3-manifolds, right?)

Answer 1

In general, any invariant of oriented manifolds is an invariant of non-orientable ones, since one can pass to the oriented double cover. Note that this cover admits an orientation-reversing diffeomorphism, so in some sense it shouldn't matter which orientation you choose. (This point should be treated with great care, especially if you are interested in maps between various groups defined on the double cover, or gradings that depend on the choice of orientation.) So in this sense the Heegaard Floer homology groups of the oriented double cover $\tilde{M}$ are invariants of your manifold $M$; in principle one could probably find a way of expressing those in terms of a Heegaard diagram for $M$. One thing to keep in mind is that Heegaard diagrams for non-orientable manifolds involve non-orientable handlebodies.

One should consider not just the Heegaard Floer groups of $\tilde{M}$, but also the action of the covering transformation. Since the covering transformation doesn't fix any points, it's not so clear how to define this, as the definition of HF involves a choice of a basepoint; see Juhász-Thurston. Presumably you have to work modulo automorphisms that result from moving the base point around. The point above about how you would define maps is also relevant here.

With regard to Rohlin invariants, you should note that the Rohlin invariant of an oriented manifold depends on a choice of spin structure. Presumably you have to choose something like a pin structure on $M$ and again try to define a Rohlin-type invariant of $\tilde{M}$. Rohlin invariants of oriented spin 3-manifolds live in $Q/16Z$; the resulting invariant would at best live in $Q/8Z$ or something like that.