Signs in Heegaard Floer

I will discuss joint work in progress with Manolescu to prove naturality of Heegaard Floer invariants away from characteristic 2. We start by using Perutz’s construction that identifies Heegaard Floer groups as Lagrangian Floer cohomology groups, bypassing Ozsvath and Szabo’s specialised construction. The main issue from this perspective is that the definition of Lagrangian Floer cohomology groups away from characteristic 2 requires making additional choices for each pair of Lagrangians, most importantly that of Pin structures following de Silva, Fukaya-Oh-Ohta-Ono, and Seidel. Ensuring that the Floer groups are independent of choices up to an unknown isomorphism then amounts to ensuring the connectedness of the space of choices required in the construction. We identify a specific space of choices, which is homotopy equivalent to real projective space, for which the strategy implemented by Juhász-Thurston-Zemke in characteristic 2 readily gives independence of the groups up to overall sign. The most laborious part of our work then amounts to removing this sign ambiguity; there is one particularly tricky commutative diagram to check.