Coherent orientations in Heegaard Floer homology

Over $\mathbb{Z}$, holomorphic disk moduli spaces in a Heegaard Floer chain complex must be oriented compatibly with gluing: the orientation of a concatenated disk class is the product determinant-line orientation of its two factors. Boundary orientations of compactified one-dimensional moduli spaces then make broken disks cancel, yielding ${\partial }^2=0$. Isolated regular disks contribute $+1$ or $-1$. Changing the chosen orientation at an intersection generator changes the corresponding row and column signs of the differential and yields an isomorphic integral chain complex. Cobordism triangle counts require compatible orientation choices as well; an overall orientation ambiguity accounts for the usual sign ambiguity of the closed invariant.