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Low-Dimensional Topology and Geometry

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Low-Dimensional Topology and Geometry SPECIAL FEATURE: INTRODUCTION Low-dimensional topology and geometry Robion C. Kirby1 Department of Mathematics, University of California, Berkeley, CA 94720 t the core of low-dimensional In 2001, Ozsváth and Szabó (6, 7) es- There are several other invariants for topology has been the classifica- tablished Heegaard Floer homology for 3-manifolds Y derived from 4-manifold A tion of knots and links in the 3-manifolds and knots in them without techniques applied to Y × R. The earliest, 3-sphere and the classification relying on a 4-dimensional theory. They Instanton Floer homology, was due of of 3- and 4-dimensional manifolds (see start with a Heegaard decomposition of Y3 course to Floer and uses the Donaldson Wikipedia for the definitions of basic to- (6, 7). This can be derived from a Morse invariants (9). Another version uses the pological terms). Beginning with the in- function f: Y → R and the index zero and Seiberg–Witten equations on Y × R, where troduction of hyperbolic geometry into one critical points of f have a neighbor- the solutions on Y are called monopoles; knots and 3-manifolds by W. Thurston in hood (the 0- and 1-handles), which is the details appear in the monograph of the late 1970s, geometric tools have be- a classical handlebody whose boundary is a Kronheimer and Mrowka (10). A third come vital to the subject. surface Σ of genus g equal to the number version, embedded contact homology Next came Freedman’s (1) classification of index one critical points (assume only (ECH), was created by Hutchings (11). of simply connected topological 4-mani- one each of critical points of index zero or A good survey of these theories can be folds in 1981 followed by the gauge theory three). Dually, the index two and three found at Wikipedia. invariants of smooth 4-manifolds intro- critical points provide another handlebody ECH requires a contact structure on Y; duced by Donaldson (2) in 1982. The with the same boundary Σ. Just how these it corresponds to the choice of an almost gauge theory invariants (2) were based two handlebodies are glued together along complex structure on Y × R. The contact on solutions to the Yang–Mills equations Σ by an element of the mapping class structure on Y is given by a differential for connections on a complex 2-plane group (the isotopy classes of diffeo- 1-form λ satisfying λ ∧ dλ > 0 everywhere bundle over the 4-manifold X4. These re- morphisms of Σ) provides all of the rich- (equivalently, a nowhere integrable 2- sults were striking, giving many smooth ness in the classification of 3-manifolds. plane field on Y). A Reeb vector field ρ on structures on many compact, closed, ori- (Mapping class groups are subtle; this is Y is defined by dλ(ρ, ·) = 0 and λ(ρ) = 1; it ented 4-manifolds. Even more striking indicated by the 100+ years needed to integrates to a flow on Y that leaves λ in- was the discovery of uncountably many prove the Poincare Conjecture, which variant. The Reeb vector field must have exotic smooth structures on ordinary 4- was finally done by Perelman (see Wiki- closed orbits, as shown by Taubes (12) in space, R4. It is possible that all compact pedia, http://en.wikipedia.org/wiki/ his proof of the Weinstein Conjecture. smooth 4-manifolds have many smooth Grigori_Perelman) using differential geo- These closed orbits, counted with multi- structures and that all noncompact smooth metric methods, not a better under- plicity, form chain groups, and again, 4-manifolds have uncountable smooth standing of the mapping class group.) pseudoholomorphic curves in Y × R,which structures. The homology of Y3 is obtained by limit on these closed orbits, give a differen- In 1994, the Seiberg–Witten equations studying the flow lines (using a Rieman- tial and then ECH. (3) were discovered, and they were a much nian metric to provide a gradient flow) Three of the four Floer homology simpler pair of equations to work with between critical points of index two and theories for Y3 (not Instanton Floer than the Yang–Mills equations. Within one. Heegaard Floer homology enhances homology) are expected to be essentially months, Taubes (4, 5) had shown that, in ordinary homology by counting pseudo- equivalent. Taubes (12) has proven that the case of a symplectic 4-manifold X4, the holomorphic curves C, where the 1- the Seiberg–Witten Floer homology is Seiberg–Witten invariants were equivalent dimensional boundary of C maps to either equivalent to ECH. In the paper of Colin to the Gromov–Witten invariants, which an index two critical point cross R or an et al. (13), the authors outline a proof that count the number of pseudoholomorphic index one critical point cross R or it limits the hat versions of ECH and Heegaard curves in X4 that belong to certain 2- on flow lines at either end of Y × R. With Floer homology are equivalent (Kutluhan dimensional homology classes. The sym- the right choice of flow lines and almost et al. have also announced a proof; refs. plectic 4-manifold has a compatible, al- complex structure on the tangent bundle 14–16). The method is to describe Y as an most-complex structure [a lifting of the of Y × R, the moduli space of pseudoho- open book; it then has a contact structure tangent bundle of X to a U(2) bundle]; the lomorphic curves is compact and 0-di- for which the Reeb vector field is posi- pseudoholomorphic curves are immersed mensional and thus, a finite number of tively transverse to the pages and tangent real surfaces whose tangent planes are points. These curves then give a boundary to the binding. The Heegaard splitting is complex lines in the U(2) bundle, and the map from one set of flow lines to another then constructed with Heegaard surface homology classes are chosen so that the whose grading differs by one and hence, equal to the union of two pages along the compact moduli space of pseudoholo- a chain complex and Heegaard Floer binding. These tools eventually lead to morphic curves is 0-dimensional and thus, homology. the proof. finite. Counting pseudoholomorphic In the first of nine papers in this Special Hutchings (11) uses ECH in a different curves is a fundamental theme underlying Feature, Lipshitz et al. (8) sketch a gener- way in his paper, which addresses the several of the papers in this alization to 3-manifolds with parametrized question of whether one symplectic 4- Special Feature. boundaries. More elaborate algebra is manifold embeds in another. Of course, The above invariants, applied to M3 × R, needed to give the desired gluing theorems the volume of the former must not be for closed, orientable 3-manifolds, M, gave when two 3-manifolds with the same pa- greater than the volume of the latter. more invariants in dimension three. Fur- rametrized boundaries are glued together. thermore, versions of the theorems for The invariants for each piece should 4-manifolds with boundaries gave in- combine to give the Heegaard Floer ho- Author contributions: R.C.K. wrote the paper. formation about links in 3-manifolds mology of the resulting 3-manifold, as in The author declares no conflict of interest. bounding surfaces in 4-manifolds. a topological quantum field theory. 1E-mail: [email protected]. www.pnas.org/cgi/doi/10.1073/pnas.1103548108 PNAS | May 17, 2011 | vol. 108 | no. 20 | 8081–8084 Downloaded by guest on September 27, 2021 There are classic results from Gromov (17) handle is <n. Therefore, the most in- curves, which limit on Z in a controlled and very recent results from McDuff (18). teresting is the case of the critical surgery, way. Note that such closed 2-forms cannot Gromov’s nonsqueezing theorem (17) when the handle of index n is attached. In exist on a homotopy 4-sphere. However, states that a symplectic 2n ball cannot be an earlier work of Bourgeois et al. (20), the notion of a 1-manifold Z, off of which symplectically embedded in a cylinder, there was found a critical surgery formula ω is nice, is an idea that works well − B2 × R2n 2 if the radius of the 2n ball for symplectic homology as additive when combined with classical Lefschetz is greater than the radius of the 2-disk. groups. The current paper studies what is fibrations. McDuff (18) considers 4-dimensional el- happening to the multiplicative structure Recall that a Lefschetz fibration, p: lipsoids and gives quite subtle necessary on symplectic homology. The authors X4 → CP1 = S2, is a complex analytic map and sufficient conditions for one ellipsoid found an explicit description of the prod- on a complex surface X, which is locally to symplectically embed in another. uct on symplectic homology after handle- a projection (the differential has complex Hutchings (11) extends McDuff’s (18) attaching in terms of certain effectively rank one) or is a Lefschetz singularity, results by using ECH capacities to give computable invariants associated with (z, w) → zw, in local coordinates. Away obstructions to the embedding of one Legendrian spheres along which the index from the (finitely many) singularities, the symplectic 4-manifold in another (which is n handles are attached. In particular, for map p is a bundle map with real 2- sharp in McDuff’s cases) (18). a 4-dimensional Weinstein symplectic dimensional fibers that are tori in the best- ECH capacities are defined for a sym- manifold, this yields an explicit combina- understood cases. Removing a neighbor- plectic 4-manifold X with a contact torial description for its symplectic ho- hood of a torus fiber and gluing it back boundary Y.
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