Heegaard Floer Homology

Heegaard Floer Homology Joshua Evan Greene Low-dimensional topology encompasses the study of man- This is a story about Heegaard Floer homology: how it ifolds in dimension four and lower. These are the shapes developed, how it has evolved, what it has taught us, and and dimensions closest to our observable experience in where it may lead. spacetime, and one of the great lessons in topology during the 20th century is that these dimensions exhibit unique Heegaard... phenomena that render them dramatically different from To set the stage, we return to the beginning of the 20th higher ones. Many outlooks and techniques have arisen in century, when Poincaréforged topology as an area of in- shaping the area, each with its own particular strengths: hy- dependent interest. In 1901, he famously and erroneously perbolic geometry, quantum algebra, foliations, geometric asserted that, as is the case for 1- and 2-manifolds, any 3- analysis,... manifold with the same ordinary homology groups as the Heegaard Floer homology, defined at the turn of the 3-dimensional sphere 푆3 is, in fact, homeomorphic to 푆3. 21st century by Peter Ozsváth and Zoltán Szabó, sits By 1905, he had vanquished his earlier claim by way of amongst these varied approaches. It consists of a power- an example that now bears his name: the Poincaréhomol- ful collection of invariants that fit into the framework ofa ogy sphere 푃3. We will describe this remarkable space in (3 + 1)-dimensional topological quantum field theory. It a moment and re-encounter it in many guises. Poincaré represents the culmination of an effort to elucidate invari- then revised his inaccurate assertion to ask whether any ants from gauge theory using symplectic geometry. The re- simply connected 3-manifold with the same homology sulting invariants have not only cast light on the structure groups as 푆3 is, in fact, homeomorphic to 푆3. We now of the earlier ones, they have opened the doors to many know this to be the case following Perelman’s resolution new ones, and they have led to tremendous breakthroughs of W. Thurston’s geometrization conjecture, and we will on the problems of low-dimensional topology. close with a folklore “proof” which is conditional on two conjectures in Heegaard Floer homology. Joshua Evan Greene is a professor of mathematics at Boston College. His email Heegaard diagrams. Poincaré described the homology 3 address is [email protected]. sphere 푃 by way of a scheme developed by Heegaard in For permission to reprint this article, please contact: his 1898 dissertation, a variation on the age-old theme [email protected]. of decomposing a complicated object into simpler pieces. In this variation, the pieces are handlebodies: a genus-푔 DOI: https://doi.org/10.1090/noti2194 JANUARY 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 19 handlebody (colloquially, a 푔-holed doughnut) is the com- To understand Figure 2, identify the circles in pairs in pact, oriented 3-manifold obtained by attaching 푔 solid such a way that like labels match. The result of the identifi- handles to a ball, as displayed in Figure 1. cation is a closed surface Σ2 with a red simple closed curve 훼1 and an orange simple closed curve 훼2. The blue seg- ments close up to a simple closed curve 훽1, and the green 푈 푉 ones to a simple closed curve 훽2. 훼1 훼2 훽1 훽2 A Heegaard diagram leads to a presentation of the ordinary homology group of the space it presents. Each pair of curves 훼푖 and 훽푗 has a well-defined intersection num- Figure 1. A pair of genus-2 handlebodies, the solid spaces ber which we can calculate by orienting and making the enclosed by the surfaces shown, along with a complete signed count of intersection points between them. Col- collection of belt circles on each. lecting these values into a 푔 × 푔 matrix, we obtain a presentation matrix 푀 for the first homology group of the A Heegaard diagram consists of a closed, oriented surface manifold. For the Heegaard diagram at hand, we obtain Σ = Σ 푔 3 2 푔 of genus , along with a pair of transverse multi- the matrix 푀 = ( −2 −1 ) of determinant 1, confirming the 3 curves 훼 = 훼1 ∪⋯∪훼푔 and 훽 = 훽1 ∪⋯∪훽푔, each composed fact that 푃 is a homology sphere. A similar procedure re- 3 of 푔 pairwise disjoint, homologically independent, simple sults in a presentation for the fundamental group 휋1(푃 ). closed curves. The data (Σ, 훼, 훽) encode a closed, oriented With a bit more effort, one finds that it has a 2-to-1 map 3-manifold 푌, as follows. We form a pair of genus-푔 han- onto the group of orientation-preserving isometries of the dlebodies 푈 and 푉 with a complete collection of belt cir- dodecahedron; hence 푃3 is not simply connected. cles surrounding the 1-handles on each. We then glue 푈 A glimpse of the construction. We can now describe and 푉 along their boundaries to Σ in such a way that the “half” of the construction of Heegaard Floer homology. map 휕푈 → Σ preserves orientation, mapping belt circles Let 푌 denote a closed, oriented 3-manifold. Present 푌 by onto the 훼 curves, and the map 휕푉 → Σ reverses orienta- means of a Heegaard diagram 퐻 = (Σ, 훼, 훽).1 Let 퐺 denote tion, mapping belt circles onto the 훽 curves. We thereby the set of 푔-tuples of points on Σ with the property that obtain a Heegaard decomposition 푈 ∪Σ 푉 of the resulting there is one point in the 푔-tuple on each curve in 훼 and space 푌. Every closed, oriented 3-manifold 푌 admits a one point on each curve in 훽. In place of the intersection Heegaard decomposition and so a Heegaard diagram. matrix 푀 that we formed to calculate 퐻 (푌), we make a 푥 1 3 new matrix whose (푖, 푗) entry is a formal sum of the inter- 푥4 section points between 훼푖 and 훽푗, signed by their intersec- 푥2 tion numbers. For instance, for the Heegaard diagram of 푃3 of Figure 2, we obtain the matrix 푥1 + 푥2 + 푥3 + 푥4 − 푥6 푥5 + 푥7 푦1 ( ) . −푦1 − 푦3 푦2 − 푦4 − 푦5 푥 1 푥3 푦 푥4 2 The monomials appearing in the formal expansion of the 푥2 푦 푦 5 determinant of this matrix correspond precisely to the set 푥 3 5 푦4 퐺. There are 19 in the example. Let 퐶퐹(퐻) denote the free 푥1 푥 abelian group generated by the set 퐺. This is the “Heegaard 푥 6 7 half” of Ozsváthand Szabó’s construction: the underlying group of a chain complex whose homology computes the Heegaard Floer homology of 푌. The group depends solely 푦 4 푦 on the combinatorics of the curve systems, and it has a 5 푥 5 (mod 2) grading coming from the signs of the monomials 푦 3 in the determinant expansion. 푦2 푦 1 ... and Floer 푥7 To describe the “Floer half,” the required differential, we must chart a course through a century of significant ad- vances that developed since the time of Heegaard and 푥6 Poincaré, passing through Morse theory, symplectic geom- Figure 2. Poincar é’s Heegaard diagram of 푃3. etry, and gauge theory. 1Strictly speaking, 퐻 should be admissible, a minor technicality. 20 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 1 Morse theory and Morse homology. In the 1930s, 휔 vanishes. Arnold and Givental conjectured that if 푀 is Morse developed an influential approach to understand- an arbitrary symplectic manifold, 퐿0 is a special class of ing smooth manifolds. Equip a smooth manifold 푀 with Lagrangian submanifold therein, and 퐿1 is a Hamiltonian a smooth, real-valued function 푓 ∶ 푀 → ℝ. The set of displacement of 퐿0 in 푀, then #(퐿0 ∩퐿1) ≥ 훽(퐿0), in direct critical points Crit(푓) consists of the points 푝 at which line with the Morse inequality. the derivative of 푓 vanishes. The function 푓 is called a In the mid-1980s, Floer attacked the Arnold–Givental Morse function if its critical points are non-degenerate. Non- conjecture by marrying the construction of the Morse com- degeneracy means that in suitable coordinates around 푝, 푓 plex with Gromov’s newly developed theory of pseudo- 2 2 2 2 takes the form 푓(푝) + 푥1 + ⋯ + 푥푘 − 푥푘+1 − ⋯ − 푥푛 for a holomorphic curves. Floer’s basic idea was to do Morse well-defined value 0 ≤ 푘 ≤ 푛 called the index of 푓 at 푝 and theory with the symplectic action functional defined on denoted ind(푝). the space of paths from one Lagrangian 퐿0 to another How many critical points must a Morse function 푓 have 퐿1 (not necessarily displacements of one another). Re- on 푀? By placing a metric 푔 on 푀 and analyzing the gra- markably, the critical points correspond to the intersec- dient flow of a suitably generic Morse function 푓, one sees tion points 퐿0 ∩ 퐿1, and, with a suitable metric in place, that 푀 admits a cell decomposition with one 푘-cell for the gradient flow trajectories to so-called pseudoholomor- each index-푘 critical point of 푓. By forming the associ- phic Whitney disks between a pair of intersection points ated cellular chain complex, we obtain the Morse inequality 푥, 푦 ∈ 퐿0 ∩ 퐿1. #Crit(푓) ≥ 훽(푀), where 훽(푀) denotes the sum of the Betti A Whitney disk from 푥 to 푦 is a smooth embedding of numbers of 푀, addressing the stated question. a disk 휙 ∶ 퐷 → 푀, 퐷 = {푧 ∈ ℂ | |푧| ≤ 1}, as in Figure In the early 1980s, Witten reinterpreted the relation- 3: namely, 휙(−푖) = 푥; 휙(푖) = 푦; 휙(푧) ∈ 퐿0 for 푧 ∈ 휕퐷, ship between Morse theory and homology by swapping Re(푧) ≤ 0; and 휙(푧) ∈ 퐿1 for 푧 ∈ 휕퐷, Re(푧) ≥ 0.

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