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.

  • $\begingroup$ I don't know anything about this field, so I can't answer your question. But I appreciate the work you put into writing it in such a clear and understandable way, so I hope someone else is able to. $\endgroup$
    – Tom Church
    Jan 25, 2018 at 6:45
  • $\begingroup$ It is an obstruction-theoretic calculation, that I have not unraveled but I refer to the explanation in Turaev's paper "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). $\endgroup$ Jan 25, 2018 at 20:03


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