I’m a beginning graduate student reading Ozsvath-Szabo’s foundational paper, Holomorphic disks and topological invariants for closed 3-manifolds. What I have trouble understanding is a formula in Lemma 2.19, a formula computing the difference of two spin^c structures. Here they claim that $ s_z(\mathbb{x})-s_z(\mathbb{y})=(deg_{D_0}(v_\mathbb{x})-deg_{D_0}(v_\mathbb{y}))Pd(\gamma_\mathbb{x}-\gamma_\mathbb{y})$ holds.

For those who are not familiar with the notations in the paper, they used an equivalent definition of spin^c structures on a closed oriented 3-manifold by Turaev: a nonvanishing vector field up to homology. They choose a pointed Heegaard diagram $ (\Sigma, \alpha, \beta, z)$ for a closed 3-maniofld $ Y$ , which is compatible with a self-indexing Morse function $ f:Y\to [0,3]$ . $ \mathbb{x}$ and $ \mathbb{y}$ are $ g$ worth collections of intersection points in $ \alpha \cap \beta$ . Then $ s_z(x)$ is represented by a vector field $ v_\mathbb{x}$ , a modification of the gradient vector field $ \nabla f$ near the geodesics passing through $ \mathbb{x}$ and $ z$ , which are denoted as $ \gamma_{\mathbb{x}}$ and $ \gamma_z$ , respectively. As the original formula trivially holds for $ \mathbb{x}=\mathbb{y}$ , assume $ \mathbb{x}\neq \mathbb{y}$ so that there is a point $ x\in \mathbb{x}-\mathbb{y}$ . Then $ D_0$ is a small neighborhood of $ x$ in $ \Sigma$ , and $ deg_{D_0}(v_\mathbb{x})-deg_{D_0}(v_\mathbb{y})$ is “the difference of the degree of $ v_\mathbb{x}|_{D_0}$ and $ v_\mathbb{y}|_{D_0}$ , thought of as a map from $ (D_0, \partial D_0)$ to $ S^2$ . (As $ v_\mathbb{x}$ and $ v_\mathbb{y}$ agrees on the boundary $ \partial D_0$ , this difference is well-defined.) And Of course $ Pd$ is the Poincare dual.

Here is my understanding:

After fixing a trivialization $ TY\cong Y\times \mathbb{R}^3$ , each nonvanishing vector fields $ v$ determine a function $ g:Y\to S^2, p\mapsto \frac{v_p}{|v_p|}$ , and vice versa. Let $ g_\mathbb{x}$ and $ g_\mathbb{y}$ be the maps corresponding to $ v_\mathbb{x}$ and $ v_{\mathbb{y}}$ , respectively. The part explaning $ s_z(\mathbb{x})-s_z(\mathbb{y})$ is a multiple of $ Pd(\gamma_\mathbb{x}-\gamma_\mathbb{y})$ is clear, so assume $ s_z(\mathbb{x})-s_z(\mathbb{y})=N(Pd(\gamma_\mathbb{x}-\gamma_{\mathbb{y}}))$ . Then $ N$ may be determined from evaluating a 2-cycle $ c$ by $ s_z(\mathbb{x})-s_z(\mathbb{y})$ and $ Pd(\gamma_\mathbb{x}-\gamma_\mathbb{y})$ : $ N=\frac{s_z(\mathbb{x})-s_z(\mathbb{y})}{Pd(\gamma_\mathbb{x}-\gamma_\mathbb{y})}$ (which makes sense for a cycle $ c$ which makes $ Pd(\gamma_\mathbb{x}-\gamma_\mathbb{y})\neq 0$ .) Then, for such a cycle $ c$ , we can find an immersed surface $ F$ realizing $ c$ , and then $ (s_z(\mathbb{x})-s_z(\mathbb{y}))[F]$ is equal to the difference $ ((g_\mathbb{x}^*-g_\mathbb{y}^*)[S^2])[S]$ , which is just $ [S^2](g_{\mathbb{x},*}[F]-g_{\mathbb{y},*}[F])=$ the difference of the degree of $ g_\mathbb{x}|_F:F\to S^2$ and $ g_\mathbb{y}|_F:F\to S^2$ . Now we can use the local degree formula to compute these two degrees. If we choose $ F$ to pass through the point $ x$ but not $ \gamma_\mathbb{y}$ (which is possible since $ \gamma_\mathbb{y}$ is simply a disjoint union of geodesics, so pushing $ F$ along $ \gamma_\mathbb{y}$ makes sense), then the only difference in the local degree formula for $ g_\mathbb{x}|_F$ and $ g_\mathbb{y}|_F$ occurs at $ x$ , which is exactly $ deg_{D_0}(v_\mathbb{x})-deg_{D_0}(v_\mathbb{y})$ times the number of intersections of $ F$ and $ \gamma_\mathbb{x}-\gamma_{y}$ . Thus the formula holds.

I reckon this is more or less the reasoning that Ozsvath and Szabo had in mind, but my concern in this argument lies on the existence of such a cycle $ c$ . When $ Y$ is a rational homology sphere so that all the 2-cycles are torsion, then clearly there exists no such $ c$ as any evaluation vanishes. Or is there a better way to avoid this issue? I guess this is a general issue in any geometric topology and not only restricted to Heegaard Floer theory, but have no idea on it. Any help would be appreciated. Thanks.