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Bordered Floer homology for sutured manifolds

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Rumen Zarev

Geometric Topology · Submitted 7 Aug 2009 · 12 references

Pith's verdict: The bridge between sutured and bordered Floer homology is real and has held up, but the analytic dictionary in §5.4 is thinner than the rest of the paper and deserves a referee's attention.

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CitationRumen Zarev. "Bordered Floer homology for sutured manifolds." arXiv:0908.1106 (2009). https://pith.science/paper/GBSBKU6R
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The paper

The paper claims

Sutured Floer homology is claimed to be recoverable from bordered invariants

The reading

Two Heegaard Floer theories for 3-manifolds with boundary have lived apart: sutured Floer homology, which needs a dividing set of curves on the boundary and says nothing about how the same manifold behaves under a different dividing set, and bordered Floer homology, which needs a parametrized boundary and glues well but carries a heavier algebraic load. Zarev proposes a common generalization. He defines bordered sutured manifolds — manifolds with corners carrying both a suture set and a parametrized piece of boundary — assembles them into a sutured cobordism category, and assigns to each such object a differential graded algebra, an $A_*$-module, or an $A_*$-bimodule, with gluing corresponding to a derived tensor product. The payoff, if the construction holds together, is that the two older theories become special cases of one functor, and that the bordered invariants of a manifold determine its sutured homology for every choice of sutures. A secondary payoff is a new proof of the surface decomposition formula for sutured Floer homology.

What carries the argument

The carrying object is the bordered sutured manifold: a sutured manifold together with an embedded ribbon graph $G(Z)$ of an arc diagram $Z$, whose neighborhood is a parametrized surface $F(Z)$ sitting in the boundary, with the arcs of $Z$ lying in the sutures and the remaining edges in the negative region. To each arc diagram the paper attaches a differential graded algebra $A(Z)$, built as a subalgebra of the strands algebra of Lipshitz–Ozsváth–Thurston, generated by idempotents together with elements $a( ho)$ recording collections of Reeb chords on $Z$ and their completions; the algebra is graded by a group $Gr(Z)$ and a reduced version $Gr(Z)$ built from a central $frac{1}{2}mathbb{Z}$ extension of $H_1(F(Z))$. To a bordered sutured manifold the paper attaches a left type $D$ structure $widehat{BSD}$, a right $A_*$-module $widehat{BSA}$, and a type $DA$ bimodule $widehat{BSDA}$, defined by counting pseudoholomorphic curves in $Sigma times [0,1] times mathbb{R}$ whose boundary punctures are labelled by Reeb chords in $Z$; curves hitting $Z$ produce algebra actions, curves hitting $partialSigma setminus Z$ are discarded, which is the structural difference from ordinary bordered Floer homology. The pairing theorem identifies $SFC$ of a glued manifold with $widehat{BSA} boxtimes_{A(Z)} widehat{BSD}$, and the bimodule gluing theorem extends this to three-piece decompositions.

Extended reading notes

Core claim

The paper's central claim is that the bordered sutured invariants define a functor. Concretely, Theorem 1 asserts that for composable decorated sutured cobordisms $(Y_1,Gamma_1,-Z_1 cup Z_2)$ and $(Y_2,Gamma_2,-Z_2 cup Z_3)$ there are graded homotopy equivalences $widehat{BSDAM}(Y_1,Gamma_1) tilde{otimes}_{A(Z_2)} widehat{BSDAM}(Y_2,Gamma_2) simeq widehat{BSDAM}(Y_1 cup Y_2, Gamma_1 cup Gamma_2)$, specializing when both outer surfaces are empty to $widehat{BSA}(Y_1,Gamma_1) tilde{otimes}_{A(Z_2)} widehat{BSDM}(Y_2,Gamma_2) simeq SFC(Y_1 cup Y_2, Gamma_1 cup Gamma_2)$. Theorem 2 asserts that the identity cobordism gets the algebra itself as its bimodule, so that $A$ and $widehat{BSDAM}$ together give a functor from the decorated sutured category to the category of differential graded algebras and $A_*$-bimodules. From this the author derives Theorem 4: for a connected 3-manifold with connected boundary, there are modules $widehat{CFA}(Gamma)$ and $widehat{CFD}(Gamma)$ depending only on the suture set such that $SFH(Y,Gamma) cong H_*(widehat{CFA}(Y) tilde{otimes} widehat{CFD}(Gamma)) cong H_*(widehat{CFA}(Gamma) tilde{otimes} widehat{CFD}(Y))$. The mechanism throughout is localization: cut the manifold into a collar piece that knows the sutures but not the 3-manifold and a complement that knows the 3-manifold but not the sutures, compute each side, and pair.

Load-bearing premise

The whole construction rests on the assumption that the analytic moduli-space machinery of bordered Floer homology — compactness, absence of boundary degenerations, and gluing — carries over unchanged to the bordered sutured setting, and on the type $DA$ bimodule formalism of a companion paper listed as in preparation.

Editorial extensions

If this is right

  • Sutured Floer homology becomes a computable functor of the bordered invariants: for a 3-manifold with connected boundary, $SFH(Y,Gamma)$ is the homology of $widehat{CFA}(Y)$ tensored with a suture-dependent module, so any bordered computation yields sutured answers for every suture set at once.
  • Juhász's surface decomposition formula, $SFH(Y',Gamma') cong bigoplus_{s text{ outer}} SFH(Y,Gamma,s)$, acquires a new proof by localizing near the decomposing surface and showing the two local bordered pieces have homotopy equivalent invariants.
  • The new invariants are defined and nontrivial for unbalanced sutured manifolds, where sutured Floer homology is not available, so the theory extends beyond the balanced setting.
  • Because the construction is functorial, any two parametrizations of the same sutured surface yield algebras that are isomorphic in the derived category, giving an explicit equivalence between their module categories.
  • Theorem 10.6 upgrades the decomposition formula to bordered sutured manifolds, so decomposition can be performed while a parametrized boundary piece is held fixed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the functoriality holds, the bordered sutured algebra $A(Z)$ is effectively a 'surface algebra' whose module category depends only on the underlying sutured surface, not on the chosen arc diagram — a statement that could be tested independently by computing $A(W_n)$ and $A(V_n)$ for the two parametrizations of the same disc-with-sutures and checking the derived equivalences explicitly.
  • The localization strategy — split a manifold into a suture-carrying collar and a suture-blind complement — looks reusable for any invariant that is local in one boundary factor, which suggests a general recipe for converting a 'gluing-friendly' theory into a 'decoration-friendly' one.
  • The paper leaves open the case of arc diagrams mixing $+$ and $-$ arcs on the same component, corresponding to cutting a Heegaard diagram along arcs crossing both $alpha$ and $beta$ circles; if that extension exists it would give a genuinely two-sided version of the theory rather than the current asymmetric one.
  • The stated connection to contact topology and to the Honda–Kazez–Matić gluing maps is asserted but not developed here, so whether the bordered sutured bimodules reproduce those contact invariants remains an open comparison.
Original abstract

We define a sutured cobordism category of surfaces with boundary and 3-manifolds with corners. In this category a sutured 3-manifold is regarded as a morphism from the empty surface to itself. In the process we define a new class of geometric objects, called bordered sutured manifolds, that generalize both sutured 3-manifolds and bordered 3-manifolds. We extend the definition of bordered Floer homology to these objects, giving a functor from a decorated version of the sutured category to A-infinity algebras, and A-infinity bimodules. As an application we give a way to recover the sutured homology SFH(Y,Gamma) of a sutured manifold from either of the bordered invariants CFA(Y) and CFD(Y) of its underlying manifold Y. A further application is a new proof of the surface decomposition formula of Juhasz.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.55 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We define a sutured cobordism category of surfaces with boundary and 3-manifolds with corners. In this category a sutured 3-manifold is regarded as a morphism from the empty surface to itself. In the process we define a new class of geometric objects, called bordered sutured manifolds, that generalize both sutured 3-manifolds and bordered 3-manifolds. We extend the definition of bordered Floer homology to these objects, giving a functor from a decorated version of the sutured category to A-infinity algebras, and A-infinity bimodules. As an application we give a way to recover the sutured homology SFH(Y,Gamma) of a sutured manifold from either of the bordered invariants CFA(Y) and CFD(Y) of its underlying manifold Y. A further application is a new proof of the surface decomposition formula of Juhasz.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We define a sutured cobordism category of surfaces with boundary and 3-manifolds with corners. contribution 0.99 0.95 0.05 asserted 1.00
s1 In this category a sutured 3-manifold is regarded as a morphism from the empty surface to itself. contribution 0.65 0.67 0.05 asserted 1.00
s2 In the process we define a new class of geometric objects, called bordered sutured manifolds, that generalize both sutured 3-manifolds and bordered 3-manifolds. contribution 1.00 0.94 0.11 asserted 1.00
s3 We extend the definition of bordered Floer homology to these objects, giving a functor from a decorated version of the sutured category to A-infinity algebras, and A-infinity bimodules. contribution 0.94 0.95 0.19 asserted 1.00
s4 As an application we give a way to recover the sutured homology SFH(Y,Gamma) of a sutured manifold from either of the bordered invariants CFA(Y) and CFD(Y) of its underlying manifold Y. result 0.59 0.95 0.33 asserted 1.00
s5 A further application is a new proof of the surface decomposition formula of Juhasz. contribution 0.55 0.93 0.11 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.62 · evidence from proof or derivation 1.00.

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Figures
Figure 1. A sutured surface (F,Λ). The sutures Λ are denoted by dots, while the positive region S+ ⊂ ∂F is colored in orange. −F1 F2 R+ R+ R− (a) Unparametrized. −F1 F2 (b) Parametrized (and with smoothed corners)
Figure 2. A sutured cobordism (Y, Γ) from a once punctured torus to a disc. The sutures Γ are colored in green, while the positive region R+ ⊂ ∂Y is shaded. sutured surface are isomorphic, and the forgetful functor Z 7→ F(Z) is an equivalence of categories. Sutured cobordisms have another, slightly different topological interpre￾tation. For a sutured cobordism (Y, Γ) from (F1,Λ1) to (F2,Λ2), we can smooth its corners, and set Γ′ = Γ ∪ S+(F1) ∪ S+(F2). This turns (Y, Γ ′ ) into a regular sutured manifold. Therefore, we can think of a sutured cobor￾dism as a sutured manifold, with two distinguis
Figure 3. Arc diagram for an annulus, and three different views of parametrization. this surgery produces no new closed components, and F is indeed a sutured surface. Alternatively, instead of a handle decomposition we can consider a Morse function on F. Whenever we talk about Morse functions, a (fixed) choice of Riemannian metric is implicit. Definition 2.3. A Z–compatible Morse function on F is a self-indexing Morse function f : F → [−1, 4], such that the following conditions hold. There are no index–0 or index–2 critical points. There are exactly k index– 1 critical points and they are all…
Figure 4. An arc diagram Z for a twice punctured torus, and its graph G(Z). vertex. An embedding of a ribbon graph into a surface will be considered orientation preserving if the ordering of the edges agrees with the positive direction on the unit tangent circle of the vertex in the surface. Definition 2.4. Let F be a sutured surface obtained from an arc diagram Z as above. The ribbon graph associated to Z is the ribbon graph G(Z) with vertices ∂Z ∪ a, and edges the components of Z \ a and the cores of the 1–handles, which we denote ei for i = 1, . . . , k. The cyclic ordering is induced from…
Figure 5. Half of a 2–handle attached along an arc. Its critical point and two incoming gradient flow lines are in the boundary. Proof. We need to give an inverse construction. Start with a bordered su￾tured diagram H, and construct a bordered sutured manifold in the following way. To Σ × [1, 2] attach 2–handles at αc i × {1}, and at βi × {2}. Finally, at αa i × {1} attach “halves of 2–handles”. These are thickened discs D2 × [0, 1] attached along an arc a × {1/2} ⊂ ∂D2 × {1/2}. (See
Figure 6. Comparison of a boundary compatible and pseudo boundary com￾patible Morse functions. Several internal critical points are given in each, with gradient flowlines, giving rise to αc and β. Such Morse functions are in 1–to–1 correspondence with compatible dia￾grams by the following construction. As usual, Σ = f −1 (3/2), while αc and β are the intersections of Σ with stable, respectively unstable, manifolds for index–1 and index–2 critical points. On the other hand, αa i is the inter￾section of Σ with the gradient flow from ei . Since the flow avoids index–1 critical points, αa is…
Figure 7. Two parametrizations of Fn. In W4 there are three Reeb chords—ρ1 from a1 to a2, ρ2 from a2 to a3, and their concatenation ρ12 from a1 to a3. The algebra splits into 4 summands. The 0– and 3–summands A(W4, 0) = hI∅i and A(W4, 3) = hI123i are trivial. The 1–summand is A(W4, 1) = hI1, I2, I3, ρ′ 1 , ρ′ 2 , ρ′ 12i. It has three idempotents and three other generators ρ ′ i = a({ρi}, ∅). It has no differential, and the only nontrivial product is ρ ′ 1 ·ρ ′ 2 = ρ ′ 12. The 2–summand A(W4, 2) = hI12, I13, I23, ρ′′ 1 , ρ′′ 2 , ρ′′ 12, ρ′′ 2 · ρ ′′ 1 i is the most interes
Figure 8. Three examples of bordered sutured manifolds (top row), and their diagrams (bottom row). Capital roman letters denote 1–handles, lower case roman letters denote intersection points, and Greek letters denote Reeb chords (always oriented upward). All non-boundary elementary regions have been shaded. (x), and I12 ·(y) = (y) (both in I(W4, 2)). There is a single region contribut￾ing to δ. It corresponds to a source S ⊲ which is a bigon from (y) to (x), with one e puncture labeled −ρ2. It contributes a2(ρ2) ⊗ (x) = ρ ′′ 2 ⊗ (x) to δ(y). Therefore, the only nontrivial term in δ is δ((y))
Figure 9. Bordered sutured decomposition of (Y, Γ) and (Y ′ , Γ ′ ). −Z2 ∼= Z1. Therefore for any s ⊂ {1, . . . , k} there is a unique generator xs ∈ G(Ht), such that I(o(x)) = I1(s) ⊗ I2(s), and l(xs) = #s. These are all the elements of G(HP ). Consider all the Spinc –structures in Spinc (T, ∂T \S × {±2}). By Poincar´e duality they are an affine space over H1(T, S × {±2}) = H1(S × [−2, 2], S × {±2}) ∼= Z, generated by an arc µ = {p} × [−2, 2] for any p ∈ S. It is easy to see that ǫ(x, y) = (l(x) − l(y)) · [µ]. Thus, for any x ∈ G(HT ), its Spinc –structure s(x) depends only on l(x). In
Figure 10. Heegaard diagrams for P and T . Here A, B, and C denote 1– handles. Notice that G(HP ) = {xP } ∼= {xT } = G(HT , sk), I(o(xP )) = I(o(xT )) = I1({1, . . . , k}) ⊗ I2(∅), and δ(xP ) = δ(xT ) = 0. Therefore BSD [(HP ) ∼= BSD [(HT , sk), and BSD [(P) ≃ BSD [(T, sk), as type D structures over A(Z). By the pairing theorem, SFC(Y ′ , Γ ′ ) ≃ BSA [(W) ⊠ BSD [(P) ≃ BSA [(W) ⊠ BSD [(T, sk) ≃ M s|T =sk SFC(Y, Γ, s). To finish the proof, we need to check that s ∈ Spinc (Y, ∂Y ) is outward to S if and only if s|T = sk. This follows from the fact that being outward to S is a local condit

Figures reproduced from arXiv:0908.1106 by the authors.

References
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38 arXiv papers cite this in 123 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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  • Next, we recall some definitions and properties of bordered-sutured Floer homology [ref] , as well as the reformulation of the bordered-sutured pairing theorem in terms of morphism complexes [ref] .

    Bordered Floer homology and incompressible surfaces · Bordered-sutured Floer homology · 0.99

  • Lipshitz–Ozsváth–Thurston's bordered Floer homology [ref] initiated this project; Zarev [ref] introduced a generalization known as bordered sutured Floer homology.

    Strands algebras and Ozsv\'ath-Szab\'o's Kauffman-states functor · Introduction · 0.90

  • Lipshitz, Oszváth, and Thurston refine Heegaard Floer homology to a bordered variant associated to a three-manifold with parametrized boundary [ref] , and Zarev [ref] further refines the invariant to an invariant of sutured manifolds with partially parameterized boundary.

    Bordered Floer homology and contact structures · Bordered sutured Floer homology · 0.93

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