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PCA ETR:INTRODUCTION FEATURE: SPECIAL

Low-dimensional and

Robion C. Kirby1 Department of , 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 for 3- Y derived from 4- A tion of and links in the 3-manifolds and knots in them without techniques applied to Y × R. The earliest, 3- and the classification relying on a 4-dimensional theory. They , 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- f: Y → R and the index zero and Seiberg–Witten 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 whose is a Kronheimer and Mrowka (10). A third come vital to the subject. Σ of 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 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 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- (the isotopy classes of diffeo- 1-form λ satisfying λ ∧ dλ > 0 everywhere over the 4-manifold X4. These re- 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 , 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 , 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 .) 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 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 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 bundle 14–16). The method is to describe Y as an most-complex structure [a lifting of the of Y × R, the of pseudoho- open book; it then has a contact structure 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 whose tangent planes are points. These curves then give a boundary to the binding. The is complex lines in the U(2) bundle, and the 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 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 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 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

8082 | www.pnas.org/cgi/doi/10.1073/pnas.1103548108 Kirby Downloaded by guest on September 27, 2021 surface cross a 2-disk lying over one com- ponent of S2 − Z and a collection of em- bedded circles (not disjoint) to which 2-handles are attached when crossing an 2 arc of Z; one then gets Fg × B over the other component of X2 − Z. When a cusp is passed, the circle to which a 2-handle is attached is changed to another circle, which intersects the first circle in exactly one point. Thus, the circles form an im- mersed daisy chain in Fg +1. In Williams’ paper (25), he introduces four natural moves and then gets a unique- Fig. 1. Trefoil . ness theorem, saying that these moves allow one to transform one simplified, purely wrinkled fibration (Williams’ modulo-two number of tori for which both four small spheres in 4-space that are name for these one-component broken bands have an odd number of twists. disjoint; one of them is shown locally as fibrations) to another. The Arf has higher-order the horizontal plane in the picture. Then, If 4-manifold invariants are defined us- generalizations as do the linking numbers each of the other 3-spheres is made to ing the combinatorics of Z in X, then the of the components of a . Conant et al. intersect this plane, each in two points and uniqueness theorems are needed to show (29) use these generalizations to give each paired by a Whitney disk. After that the invariants are invariants of the a nearly complete answer to the problem dragging around the boundaries of the 4-manifold, not just of the particular bro- of classifying the Whitney towers that Whitney disks in the horizontal plane (the ken fibration. One expected invariant aris- a link can bound in the 4-ball. A link may bounding circles are drawn in red, blue, es from counting multisections of X4 → S2, not bound disjoint surfaces, and therefore, and black in the picture), they form the which limit on Z. That is, surfaces in X the authors immerse 2-disks, each of which . It turns out that the in- whose projection back down to X2 are bounds a component of the link, and then tersection invariant of the resulting Whit- branched coverings of X2 with boundary Z try to remove pairs of intersections using ney tower has exactly three terms that projecting to p(Z). These multisections are Whitney disks. These may be immersed, correspond to the terms in the Jacobi re- similar to pseudoholomorphic curves, and so again, they try to remove inter- lation for a . Since the original which also arise as multisections of Lef- sections with Whitney disks and so on. The spheres were disjoint, this relation must schetz fibrations. This extension to all result is Whitney towers, which seem to also be introduced into the obstruction 4-manifolds follows a program of Perutz be the most natural tool for studying theory for Whitney towers and it is the key (26, 27). these questions. in translating the intersection invariant The study of topological smoothly em- Just like Milnor invariants are higher- into Milnor and Arf invariants. bedded surfaces in 4-manifolds, with no order generalizations of the linking num- Morrison and Walker (30) introduce additional geometric structure, starts with bers between two components of a link, a rather abstract notion called the blob the simple case of classifying embedded, Conant et al. (29) propose order n gen- complex. To motivate their definition, one disjoint surfaces in the 4-ball, B4, each of eralizations of Arf invariants for links can start with the familiar fundamental which bounds a given knot in the 3-sphere, (with values in certain Z2 vector group of a X with base 3 fi S . Additionally, the simplest case of this Vn). The for knots explained point x0 and describe ve independent di- 4 asks for the smallest genus surface in B , above is the case V1 =Z2, and modulo is rections in which the which a knot K bounds [this is the 4-ball the determination of the higher-order construction can be generalized. The blob genus as opposed to the minimal genus spaces Vn (only upper bounds are given); complex can be thought of as combining of a surface in S3, which is now known the paper classifies Whitney towers that all five of these generalizations: whereas π through the knot (Heegaard) Floer ho- a link can bound in the 4-ball. A link may 1(X) captures maps from the interval into mology of Ozsvath and Szabo (28)]. not be slice (i.e., not bound disjoint em- X, the blob complex B*(M ; C) provides There are some knots other than the bedded disks but immersed disks always a notion of maps from any manifold M unknot that bound 2-disks in B4, called exist in the 4-ball). The authors try to re- into an n C. slice knots, but the , the sim- move pairs of intersections between such Recall that the fundamental group π plest actual knot, can be shown not to be disks using Whitney disks. These may be 1(X, x0) is given by homotopy classes of slice by use of the most fundamental of immersed, and therefore, they try to re- loops in X based at x0, thought of as maps knot invariants, the Arf invariant, whose move intersections with Whitney disks and of the interval [0, 1] into X that send the α β values lie in the group of order two. Any so on. If n layers of such Whitney disks are endpoints to x0. Two loops, and , can be knot in S3 bounds a (ori- chosen, the resulting 2-complex is called multiplied by first traversing α and then β. ented and connected), and the trefoil a Whitney tower of order n. It has an in- This multiplication is associative but knot can be drawn so as to exhibit this tersection invariant, reflecting all order n only up to homotopy. (In this case, the surface, a punctured torus, as in Fig. 1. Milnor and Arf invariants of the link on images of the loops coincide, but they are The punctured torus is a thickened figure the boundary of the Whitney tower. The parametrized differently.) 8 from ref. 28, and each (darker) circle of in the paper is based on In the first generalization of the funda- the figure 8 from ref. 28 has a neighbor- the result that this intersection invariant mental group, homotopy classes of loops hood that is a band with one full left- vanishes if and only if the construction can are replaced with actual loops. Loops handed twist. In general, a knot bounds be continued one more step [i.e., the given (based at x0) can still be multiplied, but a surface, which is a boundary connected link bounds, in fact, a Whitney tower of this multiplication will no longer be asso- sum of g (the genus) tori. The circles of the order (n + 1)]. ciative. It will, however, be associative up generalized figure 8 from ref. 28 (a bou- The picture on the cover page of this to deformations, and these deformations quet of circles) all lie in bands with twists. volume is related to the main new ma- can be incorporated into a more compli- The Arf invariant of the knot is the neuver on Whitney towers. It starts with cated structure. It turns out to be useful to

Kirby PNAS | May 17, 2011 | vol. 108 | no. 20 | 8083 Downloaded by guest on September 27, 2021 consider deformations between deforma- a standard example of an n-category. and an A∞ n-category C to produce a fi fi n tions and so on ad in nitum. The resulting Adding in the rst generalization (dis- chain complex B*(M ; C), which can be structure incorporating loops, multiplica- tinguishing deformation equivalent maps) thought of as the chains on the space of tion and iterated deformations is called an gives an example of an A∞ n-category. maps from M to C. A∞ space [whose connected components One can think of an n-category as an al- Lipshitz et al. (8) show that the Hee- π happen to form a group, namely 1(X, x0)]. gebraic structure with properties that gaard–Floer spaces of 3-manifolds satisfy Bn In the second generalization, paths are mimic the ways in which maps from to a gluing law that involves taking a tensor considered instead of loops, the endpoints X can be glued together. product over a certain A∞ algebra associ- of the interval [0, 1] can map to different In the fourth generalization, n-balls are points of X. Paths can be multiplied but replaced with arbitrary n-manifolds Mn ated to a 2-manifold. The blob complex only if the final endpoint of the first and maps of Mn into X. Since Mn can be also has an A∞ tensor product gluing law coincides with the initial endpoint of carved up into copies of the standard ball like this, and therefore, it is natural to ask the second path. The resulting algebraic Bn, these mapping spaces are in some if there is some A∞ three-category HF n 3 3 structure is a category (with invertible sense determined by the maps from B to such that HF*(M ) ≅ HF*(M ;HF). morphisms) rather than a group. X of the previous paragraph. Put another Another possible application for the In the third generalization, loops are way, maps from Bn to X are a local in- blob complex to 4-manifolds concerns replaced with maps of n-dimensional variant that can be integrated to construct Khovanov homology. Modulo some sign π n spheres into X. This old notion, n(X, x0), the global invariant of maps from M to X. issues, one should be able to construct fi fi goes back to the 1930s. It can be combined In the fth and nal generalization, note a 4-category K from Khovanov homology with the second generalization to consider that the integration procedure of the pre- and then associate the chain complex n-dimensional paths (that is, maps of the n vious paragraph works just as well if the B (W4; K) to a 4-manifold W. ball Bn into X). These maps can be mul- maps from Bn to X are replaced by maps * ∞ tiplied (i.e., glued together) in various to any n-category or A n-category. This ACKNOWLEDGMENTS. This work was supported ways when their boundaries partially co- is essentially what the blob complex is: in part by US National Science Foundation Grant n incide. The resulting algebraic structure is a way of combining an n-manifold M EMSW21-RTG.

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