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.

"Euler structures, nonsingular vector fields, and torsions of Reidemeister type", page 13 (it's a more general setup: your scenario is $m=3$ and uses the Heegaard decomposition). $\endgroup$ – Chris Gerig Jan 25 '18 at 20:03