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Ciprian Manolescu A combinatorial approach to Heegaard Floer invariants Ciprian Manolescu UCLA July 5, 2012 Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 1 / 34 Motivation (4D) Much insight into smooth 4-manifolds comes from PDE techniques: gauge theory and symplectic geometry. Donaldson (1980’s): used the Yang-Mills equations to get constraints on the intersection forms of smooth 4-manifolds, etc. Solution counts −→ Donaldson invariants, able to detect exotic smooth structures on closed 4-manifolds. Seiberg-Witten (1994): discovered the monopole equations, which can replace Yang-Mills for most applications, and are easier to study. Solution counts −→ Seiberg-Witten invariants. Ozsv´ath-Szab´o (2000’s): developed Heegaard Floer theory, based on counts of holomorphic curves in symplectic manifolds. Their mixed HF invariants of 4-manifolds are conjecturally the same as the Seiberg-Witten invariants, and can be used for the same applications (in particular, to detect exotic smooth structures). Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 2 / 34 Motivation (3D) All three theories (Yang-Mills, Seiberg-Witten, Heegaard-Floer) also produce invariants of closed 3-manifolds, in the form of graded Abelian groups called Floer homologies. These have applications of their own, e.g.: What is the minimal genus of a surface representing a given homology class in a 3-manifold? (Kronheimer-Mrowka, Ozsv´ath-Szab´o) Does a given 3-manifold fiber over the circle? (Ghiggini, Ni) In dimension 3, the Heegaard-Floer and Seiberg-Witten Floer homologies are known to be isomorphic: work of Kutluhan-Lee-Taubes and Colin-Ghiggini-Honda, based on the relation to Hutchings’s ECH. There also exist Floer homologies for knots (or links) K ⊂ S3. Applications: knot genus, fiberedness of the knot complement, concordance, unknotting number, etc. Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 3 / 34 Motivating problem All these gauge-theoretic or symplectic-geometric invariants are defined by counting solutions of nonlinear elliptic PDE’s. As such, they are very difficult to compute. Question Are the (Yang-Mills, Seiberg-Witten, Heegaard-Floer) invariants of knots, 3-manifolds, and 4-manifolds algorithmically computable? In other words, do they admit combinatorial definitions? The answer is now Yes for the Heegaard-Floer invariants (mod 2). This is based on joint work with / work of (various subsets of) : Robert Lipshitz, Peter Ozsv´ath, Jacob Rasmussen, Sucharit Sarkar, Andr´as Stipsicz, Zoltan Szab´o,Dylan Thurston, Jiajun Wang. Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 4 / 34 Outline 1 Summary of Heegaard Floer theory 2 Algorithms for the HF invariants of knots and links in S3 3 Presenting 3-manifold and 4-manifolds in terms of links in S3 4 From HF link invariants to HF invariants of 3- and 4-manifolds 5 An algorithm for the HF invariants of 3- and 4-manifolds 6 Open problems Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 5 / 34 Summary of Heegaard Floer theory For simplicity, we work over the field F = Z/2Z. Ozsv´ath-Szab´o(2000) Y 3 closed, oriented 3-manifold → HF−(Y ), HF (Y ), HF +(Y ) = modules over the power series ring F[[U]] (variants of Heegaardc Floer homology). Start with a marked Heegaard diagram for Y : Σ = surface of genus g α = {α1,...,αg } collection of disjoint, homologically linearly independent, simple closed curves on Σ, specifying a handlebody Uα β = {β1,...,βg } a similar collection, specifying Uβ a basepoint z ∈ Σ −∪αi −∪βi such that Y = Uα ∪Σ Uβ. Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 6 / 34 A marked Heegaard diagram Y 3 Uβ β2 z Σ β1 β3 α1 α2 α3 Uα g α 3 α Uα =Σ ∪ [ Di ∪ B , ∂Di = αi i=1 g β 3 β Uβ =Σ ∪ [ Di ∪ B , ∂Di = βi . i=1 Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 7 / 34 Heegaard Floer homology Consider the tori Tα = α1 ×···× αg Tβ = β1 ×···× βg g inside the symmetric product Sym (Σ) = (Σ ×···× Σ)/Sg , viewed as a symplectic manifold. The Heegaard Floer complex CF−(Y ) is freely generated (over F[[U]]) by intersection points x = {x1,..., xg }∈ Tα ∩ Tβ with xi ∈ αi ∩ βj ⊂ Σ. The differential is given by counting pseudo- g holomorphic disks in Sym (Σ) with boundaries on Tα, Tβ. These are solutions to the nonlinear Cauchy-Riemann equations, and depend on the choice of a generic family of almost complex structures J on the symmetric product. Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 8 / 34 z × Symg−1(Σ) Tβ y Tα φ x n (φ) ∂x = X X n(φ, J) · U z y, T T y∈ α∩ β φ∈π2(x,y) where π2(x, y) is the space of relative homology classes of Whitney disks from x to y, and n(φ, J) ∈ F = {0, 1} is the count of holomorphic disks in the class φ. Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 9 / 34 The variable U keeps track of the quantity nz (φ), the intersection number between φ and the divisor {z}× Symg−1(Σ). One can show that ∂2 = 0. We define: − − HF = H∗(CF ); − HF = H∗(CF /(U = 0)); c + −1 − − HF = H∗(U CF /CF ). Ozsv´ath and Szab´oproved that these are invariants of the 3-manifold Y . Typically, HF has less information than HF− and HF + (although it suffices forc most 3D applications). For example, HF−(S3)= F[[U]], HF (S3)= F, HF +(S3)= F[U−1]. c All three variants (◦ =+, −, hat) split according to Spinc-structures on Y : ◦ ◦ HF (Y )= M HF (Y , s) s∈Spinc(Y ) Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 10 / 34 Heegaard Floer theory as a TQFT The three variants of Heegaard Floer homology are functorial under Spinc-decorated cobordisms. (Y0, s0) (W , s) (Y1, s1) ◦ ◦ ◦ FW ,s : HF (Y0, s0) −→ HF (Y1, s1) The maps are defined by counting pseudo-holomorphic triangles in the symmetric product, with boundaries on three different tori. Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 11 / 34 Mixed HF invariants of 4-manifolds (Ozsv´ath-Szab´o, 2001) 4 + X = closed, oriented, smooth 4-manifold with b2 (X ) > 1 N = admissible cut + b (W±) > 0 4 B4 2 B W− W+ δH1(N)=0 ⊂ H2(X ) N F − W− − 1 ∈ F[[U]] = HF −(S3) HF (N) HF red(N) F + W+ − HF +(N) HF +(S3) = F[U 1] ΨX ,s ∈ {0, 1} detects exotic smooth structures; conjecturally = SWX ,s Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 12 / 34 Going back to a marked Heegaard diagram (Σ, α, β, z), if one specifies another basepoint w, this gives rise to a knot K ⊂ Y . Ozsv´ath-Szab´o, Rasmussen (2003) Counting pseudo-holomorphic disks and keeping track of the quantities nz (φ) and nw (φ) in various ways yields different versions of knot Floer homology HFK ◦(Y , K). Y 3 K z Σ w One can also break the knot (or link) into more segments, by using more basepoints. Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 13 / 34 Simplest version of knot Floer homology: HFK(S 3, K) We count only pseudo-holomorphic disks with nz (φ)= nw (φ) = 0: 3 3 HFK(S , K)= M HFK m(S , K, s) m,s∈Z Its Euler characteristic is the Alexander polynomial of the knot: m s 3 X (−1) q · rk HFK m(S , K, s)=∆K (q). m,s∈Z The genus of a knot g(K) = min{g |∃ embedded, oriented,Σ2 ⊂ S3 of genus g, ∂Σ= K} can be read from HFK (Ozsv´ath-Szab´o, 2004): 3 g(K) = max{s ≥ 0 |∃ m, HFK m(S , K, s) = 0}. Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 14 / 34 More about HFK(S 3, K) In particular: 3 ∼ F K is the unknot (g(K) = 0) ⇐⇒ HFK(S , K) = (0,0). By a result of Ghiggini (2006), knot Floer homology has enough information to also detect the right-handed trefoil, the left-handed trefoil, and the figure-eight knot. Ni (2006) showed that S3 \ K fibers over the circle if and only if 3 ⊕mHFK m(S , K, g(K)) =∼ F. Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 15 / 34 Algorithms for computing knot (and link) Floer homology Theorem (M.-Ozsv´ath-Sarkar, 2006) All variants of Heegaard Floer homology for links L ⊂ S3 are algorithmically computable. Every link in S3 admits a grid diagram; that is, an n-by-n grid in the plane with O and X markings inside such that: Each row and each column contains exactly one X and one O; As we trace the vertical and horizontal segments between O’s and X ’s (verticals on top), we see the link L. Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 16 / 34 A grid diagram for the trefoil Grid diagrams are particular examples of Heegaard diagrams: if we identify the opposite sides to get a torus, we let: α = horizontal circles, β = vertical circles; the X ’s and O’s are the basepoints. Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 17 / 34 Generators The generators x = {x1,..., xn} are n-tuples of points on the grid (one on each vertical and horizontal circle). Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 18 / 34 Generators The generators x = {x1,..., xn} are n-tuples of points on the grid (one on each vertical and horizontal circle). Ciprian Manolescu (UCLA) Combinatorial HF Theory July 5, 2012 19 / 34 Differentials In a grid diagram, isolated pseudo-holomorphic disks are in 1-to-1 correspondence to empty rectangles on the grid torus (that is, having no red or blue dots inside).
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