Cut-System Complex in Sutured Manifolds

• Cut-System Complex is a 2-dimensional cell complex built from cut-systems and elementary handleslides in a sutured compression body.
• It incorporates six basic 2-cell handleslide loops, such as slide-triangles and squares, that ensure the complex is connected and simply connected.
• The complex serves as a combinatorial backbone to extend tight Heegaard invariants uniquely to strong invariants in Floer homology.

Searching arXiv for the cited sutured Floer / Heegaard invariant background papers. The cut-system complex, in the sutured-manifold sense, is the $2$-dimensional cell complex associated to a sutured compression body whose vertices are cut-systems, whose edges are elementary handleslides, and whose $2$-cells are attached along six basic handleslide loops. For a sutured compression body $C(\delta)$, the resulting complex $Y_2(C(\delta))$ is connected and simply connected after these six classes of $2$-cells are added. In the same framework, tight Heegaard invariants admit unique extensions to strong Heegaard invariants, yielding a combinatorial route to naturality results for Floer homology theories associated to sutured manifolds (Qin, 6 Aug 2025).

1. Sutured compression bodies and cut-systems

Fix a compact oriented surface $\Sigma$, possibly disconnected, such that every boundary component is nonempty, and assume there are no special disks. A collection

$\delta=(\delta_1,\dots,\delta_k)$

of $k$ disjoint simple closed curves in $\operatorname{Int}(\Sigma)$ is an attaching set if each component of $\Sigma\setminus \delta$ meets $2$0 (Qin, 6 Aug 2025).

The associated sutured compression body $2$1 is obtained by starting with $2$2, attaching $2$3-handles to $2$4 along $2$5, and taking the suture

$2$6

Its lower boundary is

$2$7

and its upper boundary is

$2$8

A meridian curve in $2$9 is a curve bounding a compressing disk in $C(\delta)$0. The collection $C(\delta)$1 is a cut-system of $C(\delta)$2 if $C(\delta)$3 equals the number of $C(\delta)$4-handles and cutting $C(\delta)$5 along the compressing disks for $C(\delta)$6 produces a product sutured manifold $C(\delta)$7. Equivalently, $C(\delta)$8 is a cut-system if it is a maximal set of disjoint meridians and $C(\delta)$9 is a union of punctured spheres each meeting $Y_2(C(\delta))$0.

This definition places the cut-system at the level of maximal compression data for the sutured compression body. A plausible implication is that the complex built from such data records not merely existence of compressions, but the full combinatorics of moving between maximal compression configurations by handleslides.

2. The cell complex $Y_2(C(\delta))$1

The cut-system complex begins with a graph $Y_2(C(\delta))$2. Its vertices are all cut-systems

$Y_2(C(\delta))$3

where $Y_2(C(\delta))$4. Its edges are elementary handleslides: there is an edge $Y_2(C(\delta))$5 whenever

$Y_2(C(\delta))$6

and $Y_2(C(\delta))$7 is obtained from $Y_2(C(\delta))$8 by sliding over one of the other curves along an embedded arc disjoint from the remaining $Y_2(C(\delta))$9's (Qin, 6 Aug 2025).

Thus $2$0 is the graph of cut-systems and single handleslides. The full complex $2$1 is obtained by attaching $2$2-cells along six distinguished handleslide loops.

The construction is explicitly $2$3-dimensional: it does not attempt to encode all higher homotopy data directly. Instead, it identifies a finite list of local handleslide relations sufficient to force simple connectivity. This is the essential structural content of the complex.

3. The six handleslide $2$4-cells

Let $2$5 denote a fixed collection of $2$6 curves. The six basic loops in $2$7 are the following, each filled by a $2$8-cell (Qin, 6 Aug 2025).

Loop type	Configuration	Cell attached along
Slide-triangle	$2$9 cobound a pair of pants disjoint from $\Sigma$0	A $\Sigma$1-cycle of slides
Type I square	Sliding $\Sigma$2 is independent of sliding $\Sigma$3	A commutative square
Type II square	$\Sigma$4 and $\Sigma$5 both slide over $\Sigma$6 from opposite sides	A commutative square
Type III square	Sliding $\Sigma$7 over either $\Sigma$8 or $\Sigma$9 gives the same result up to homotopy	A square
Type IV square	$\delta=(\delta_1,\dots,\delta_k)$0 slides over $\delta=(\delta_1,\dots,\delta_k)$1 in two distinct commuting ways	A square
Slide pentagon	Five curves bound a disk with five marked boundary intervals	A $\delta=(\delta_1,\dots,\delta_k)$2-cycle

The slide-triangle is written explicitly as

$\delta=(\delta_1,\dots,\delta_k)$3

For the square and pentagon relations, the paper identifies Type II, Type III, Type IV, and the pentagon with the corresponding configurations labeled as cases $\delta=(\delta_1,\dots,\delta_k)$4, $\delta=(\delta_1,\dots,\delta_k)$5, $\delta=(\delta_1,\dots,\delta_k)$6, and $\delta=(\delta_1,\dots,\delta_k)$7 of Juhász–Thurston–Zemke (Qin, 6 Aug 2025).

The role of these $\delta=(\delta_1,\dots,\delta_k)$8-cells is local but decisive: they kill precisely the elementary cycles generated by compatible or competing handleslide operations. This suggests that the global topology of the cut-system complex is controlled by a small, finite list of move relations.

4. Connectivity and simple connectivity

The main theorem states:

Let $\delta=(\delta_1,\dots,\delta_k)$9 be any sutured compression body and let $k$0 be the $k$1-dimensional cell complex obtained by starting from the $k$2-skeleton of cut-systems and elementary handleslides and then attaching the six kinds of $k$3-cells listed above. Then $k$4 is connected and simply connected (Qin, 6 Aug 2025).

The proof has three stated components. First, connectivity is immediate from the fact that any two compression-equivalent attaching sets differ by a sequence of handleslides, cited to Juhász–Thurston–Zemke, Lemma 2.11. Second, simple connectivity is reduced to an adaptation of Wajnryb’s complex $k$5, whose vertices are cut-systems and whose edges are simple moves, meaning replacement of one $k$6-curve by another disjoint one. That complex carries triangular and square $k$7-cells and is shown to be simply connected by induction on the number of curves. Third, a minimal resolution argument replaces each simple-move edge in $k$8 by a canonical path of handleslides in $k$9, unique up to the $\operatorname{Int}(\Sigma)$0-cell relations already listed. Each $\operatorname{Int}(\Sigma)$1-cell in $\operatorname{Int}(\Sigma)$2 is then expressed as a composition of slide-triangles, squares, and pentagons in $\operatorname{Int}(\Sigma)$3 (Qin, 6 Aug 2025).

The theorem gives a presentation-level control of handleslide monodromy. In practical terms, any loop in the handleslide graph can be reduced using the six basic $\operatorname{Int}(\Sigma)$4-cell relations. A plausible implication is that the complex functions as a coherence object for diagrammatic moves in sutured Floer theory.

5. Tight and strong Heegaard invariants

In the Heegaard-theoretic application, Juhász–Thurston–Zemke package a Floer-type assignment to each sutured Heegaard diagram as a functor on a graph $\operatorname{Int}(\Sigma)$5 of diagrams and moves, including $\operatorname{Int}(\Sigma)$6-equivalences, stabilizations, and diffeomorphisms. Over $\operatorname{Int}(\Sigma)$7, the $\operatorname{Int}(\Sigma)$8-equivalences suffice, but over $\operatorname{Int}(\Sigma)$9 one must work with $\Sigma\setminus \delta$0-handleslides to avoid sign-ambiguities. Replacing equivalences by handleslides yields a subgraph $\Sigma\setminus \delta$1 (Qin, 6 Aug 2025).

A tight Heegaard invariant is a functor

$\Sigma\setminus \delta$2

that commutes around exactly the six slide-loops above, together with stabilization-slides. The simple connectivity of $\Sigma\setminus \delta$3 implies that any such tight invariant extends uniquely to a strong invariant on the full graph $\Sigma\setminus \delta$4.

The consequence recorded in the paper is that sutured Floer homology, link Floer homology, and multi-pointed Heegaard Floer homology admit unique strong refinements, proving their naturality over $\Sigma\setminus \delta$5; equivalently, there is no monodromy around any loop of Heegaard moves. Within this framework, the cut-system complex provides the combinatorial backbone for existence and uniqueness of strong Heegaard invariants.

A common misconception is that naturality in Floer theory is purely formal once diagrammatic moves are known. The result here is more specific: the passage from tight to strong invariants depends on the simple connectivity of the handleslide-based cut-system complex and on the six explicit slide relations.

6. Terminological scope and adjacent usages

The term “cut-system complex” is not uniform across the literature. In graph theory, “total cut complexes” and “cut complexes” denote simplicial complexes associated to a finite graph $\Sigma\setminus \delta$6, written $\Sigma\setminus \delta$7 and $\Sigma\setminus \delta$8, with faces determined by complements containing independent $\Sigma\setminus \delta$9-sets or disconnected induced $2$00-vertex subgraphs. For grid graphs $2$01 and $2$02, these complexes are studied via Alexander duality, shellability, and discrete Morse theory, and theorems are given describing wedge-of-spheres decompositions and shellability in the $2$03 and $2$04 cases (Chandrakar et al., 2024).

In convex optimization, the “complex cut polytope” is a different object again: $2$05 the convex hull of rank-one Hermitian matrices determined by $2$06th roots of unity. That literature concerns valid inequalities, semidefinite liftings, and relaxations for problems such as $2$07, MIMO detection, and angular synchronization (Sinjorgo et al., 2024).

These usages are mathematically unrelated to the sutured-manifold cut-system complex except at the level of nomenclature. This suggests that, in the present topic, “cut-system complex” should be understood specifically as a handleslide complex for cut-systems of a sutured compression body, not as a graph-theoretic simplicial complex or a polyhedral object.