Sutured Floer chain complexes and homology notation

For a balanced sutured manifold $(M,\gamma )$, a balanced sutured Heegaard diagram is a tuple $\mathcal{H}=(\mathrm{\Sigma },\mathit{\alpha },\mathit{\beta })$ where $\mathrm{\Sigma }$ is a compact oriented surface with boundary and each of $\mathit{\alpha },\mathit{\beta }$ is a finite family of pairwise disjoint simple closed curves in $\mathrm{int}\mathrm{\Sigma }$. It defines $M$ by attaching $3$-dimensional $2$-handles to $\mathrm{\Sigma }\times [0,1]$ along $\mathit{\alpha }\times \{0\}$ and $\mathit{\beta }\times \{1\}$, with sutures $\mathrm{\partial }\mathrm{\Sigma }\times \{1\mathrm{/}2\}$ on $\gamma =\mathrm{\partial }\mathrm{\Sigma }\times [0,1]$. The diagram is balanced when the two families have equal cardinality $d$ and the natural maps from components of $\mathrm{\partial }\mathrm{\Sigma }$ to components of $\mathrm{\Sigma }\setminus \bigcup \mathit{\alpha }$ and $\mathrm{\Sigma }\setminus \bigcup \mathit{\beta }$ are both surjective. A domain is an integral two-chain generated by the closures of regions of $\mathrm{\Sigma }\setminus (\bigcup \mathit{\alpha }\cup \bigcup \mathit{\beta })$ disjoint from $\mathrm{\partial }\mathrm{\Sigma }$. A periodic domain is a domain whose boundary is an integral sum of full $\alpha$- and $\beta$-curves. The diagram is admissible when each nonzero periodic domain has at least one positive and one negative region coefficient.

The symmetric product ${\mathrm{Sym}}^d(\mathrm{\Sigma })={\mathrm{\Sigma }}^d\mathrm{/}{S}_d$ is the space of unordered $d$-tuples of points. The product tori ${\mathbb{T}}_{\alpha },{\mathbb{T}}_{\beta }$ are the images of the products of the $d$ attaching curves in this symmetric product. Require curves from different families to intersect transversely. A generator is a point of ${\mathbb{T}}_{\alpha }\cap {\mathbb{T}}_{\beta }$, equivalently a collection of $d$ transverse intersection points, one on each attaching curve in each family. A Whitney disk class from generator $\mathbf{x}$ to generator $\mathbf{y}$ is a relative homotopy class of maps of a disk into ${\mathrm{Sym}}^d(\mathrm{\Sigma })$ whose two boundary arcs lie on the product tori and whose endpoints are $\mathbf{x},\mathbf{y}$; ${\pi }_2(\mathbf{x},\mathbf{y})$ denotes the set of these classes. For $d>0$, fix a smooth path $J_t$ of almost complex structures on ${\mathrm{Sym}}^d(\mathrm{\Sigma })$ for which the product tori are totally real and which is regular for the disk classes used in the differential and its squared-differential argument. Here an almost complex structure is a smooth bundle endomorphism $J$ with $J^2=-\mathrm{id}$, and regularity means that the linearized Cauchy--Riemann operator at every nonconstant holomorphic strip in those classes is surjective; in particular impose this for Maslov indices one and two. The counts below use such regular data, obtained by generic perturbation within the sutured Floer analytic setup. A representative of a Whitney disk class is a smooth strip $u:\mathbb{R}\times [0,1]\to {\mathrm{Sym}}^d(\mathrm{\Sigma })$ satisfying ${\mathrm{\partial }}_{s}u+J_t(u){\mathrm{\partial }}_{t}u=0$, with $u(s,0)\in {\mathbb{T}}_{\alpha }$, $u(s,1)\in {\mathbb{T}}_{\beta }$, and limits $\mathbf{x},\mathbf{y}$ as $s\to -\mathrm{\infty },+\mathrm{\infty }$. Its Maslov index is the Fredholm index of the linearized Cauchy--Riemann operator at a representative, and ${\mathcal{M}}_J(\varphi )$ denotes the space of representatives of class $\varphi$. Translation of the real coordinate gives the $\mathbb{R}$ action on this moduli space.

When $d=0$, ${\mathrm{Sym}}^0(\mathrm{\Sigma })$ is a point, both product tori are that point, and the unique generator is the empty collection.

For a fixed admissible diagram and the chosen regular data, the sutured Floer disk-count module $SFC(\mathcal{H};R)$ is the free $R$-module on its generators, where $R={\mathbb{F}}_2$ or $\mathbb{Z}$. For $R={\mathbb{F}}_2$, its formal differential counts translation classes modulo two: $\mathrm{\partial }\mathbf{x}={\sum }_{\mathbf{y}}{\sum }_{\begin{array}{c}\varphi \in {\pi }_2(\mathbf{x},\mathbf{y})\\ \mu (\varphi )=1\end{array}}{\mathrm{\#}}_2({\mathcal{M}}_J(\varphi )\mathrm{/}\mathbb{R})\mathbf{y}\mathrm{.}$ For $R=\mathbb{Z}$, use a coherent orientation system, which assigns orientations to determinant lines of the linearized disk operators compatibly with gluing, and replace ${\mathrm{\#}}_2$ by the signed count ${\mathrm{\#}}_{\mathfrak{o}}$. The homology of this diagram-level differential is denoted $H_{\ast }(SFC(\mathcal{H};R))$ once the moduli counts are finite and the differential squares to zero. When $d=0$ the unique empty generator has zero differential. Once admissible diagram-change maps are proved to be chain homotopy equivalences, their common homology is denoted $SFH(M,\gamma ;R)$. This definition fixes the chain-level and coefficient conventions; it does not assert analytic compactness, gluing, invariance, or surface-decomposition results.

Balanced and taut sutured manifolds, product regions, and horizontal surfaces

Whitney disks, product tori, Maslov index, and holomorphic disk counts

Juhasz, Floer homology and surface decompositions, §§2–3

Ni, Knot Floer homology detects fibred knots, §2.2