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Low-dimensional topology and geometry

  1. Robion C. Kirby1
  1. Department of Mathematics, University of California, Berkeley, CA 94720

At the core of low-dimensional topology has been the classification of knots and links in the 3-sphere and the classification of 3- and 4-dimensional manifolds (see Wikipedia for the definitions of basic topological terms). Beginning with the introduction of hyperbolic geometry into knots and 3-manifolds by W. Thurston in the late 1970s, geometric tools have become vital to the subject.

Next came Freedman's (1) classification of simply connected topological 4-manifolds in 1981 followed by the gauge theory invariants of smooth 4-manifolds introduced by Donaldson (2) in 1982. The gauge theory invariants (2) were based on solutions to the Yang–Mills equations for connections on a complex 2-plane bundle over the 4-manifold X4. These results were striking, giving many smooth structures on many compact, closed, oriented 4-manifolds. Even more striking was the discovery of uncountably many exotic smooth structures on ordinary 4-space, R4. It is possible that all compact smooth 4-manifolds have many smooth structures and that all noncompact smooth 4-manifolds have uncountable smooth structures.

In 1994, the Seiberg–Witten equations (3) were discovered, and they were a much simpler pair of equations to work with than the Yang–Mills equations. Within months, Taubes (4, 5) had shown that, in the case of a symplectic 4-manifold X4, the Seiberg–Witten invariants were equivalent to the Gromov–Witten invariants, which count the number of pseudoholomorphic curves in X4 that belong to certain 2-dimensional homology classes. The symplectic 4-manifold has a compatible, almost-complex structure [a lifting of the tangent bundle of X to a U(2) bundle]; the pseudoholomorphic curves are immersed real surfaces whose tangent planes are complex lines in the U(2) bundle, and the homology classes are chosen so that the compact moduli space of pseudoholomorphic curves is 0-dimensional and thus, finite. Counting pseudoholomorphic curves is a fundamental theme underlying several of the papers in this Special Feature.

The above invariants, applied to M3 × R, for closed, orientable 3-manifolds, M, gave more invariants in dimension three. Furthermore, versions of the theorems for 4-manifolds with boundaries gave information about links in 3-manifolds bounding surfaces in 4-manifolds.

In 2001, Ozsváth and Szabó (6, 7) established Heegaard Floer homology for 3-manifolds and knots in them without relying on a 4-dimensional theory. They start with a Heegaard decomposition of Y3 (6, 7). This can be derived from a Morse function f: YR and the index zero and one critical points of f have a neighborhood (the 0- and 1-handles), which is a classical handlebody whose boundary is a surface Σ of genus g equal to the number of index one critical points (assume only one each of critical points of index zero or three). Dually, the index two and three critical points provide another handlebody with the same boundary Σ. Just how these two handlebodies are glued together along Σ by an element of the mapping class group (the isotopy classes of diffeomorphisms of Σ) provides all of the richness in the classification of 3-manifolds. (Mapping class groups are subtle; this is indicated by the 100+ years needed to prove the Poincare Conjecture, which was finally done by Perelman (see Wikipedia, http://www.danielhellerman.com/wiki/Grigori_Perelman) using differential geometric methods, not a better understanding of the mapping class group.)

The homology of Y3 is obtained by studying the flow lines (using a Riemannian metric to provide a gradient flow) between critical points of index two and one. Heegaard Floer homology enhances ordinary homology by counting pseudoholomorphic curves C, where the 1-dimensional boundary of C maps to either an index two critical point cross R or an index one critical point cross R or it limits on flow lines at either end of Y × R. With the right choice of flow lines and almost complex structure on the tangent bundle of Y × R, the moduli space of pseudoholomorphic curves is compact and 0-dimensional and thus, a finite number of points. These curves then give a boundary map from one set of flow lines to another whose grading differs by one and hence, a chain complex and Heegaard Floer homology.

In the first of nine papers in this Special Feature, Lipshitz et al. (8) sketch a generalization to 3-manifolds with parametrized boundaries. More elaborate algebra is needed to give the desired gluing theorems when two 3-manifolds with the same parametrized boundaries are glued together. The invariants for each piece should combine to give the Heegaard Floer homology of the resulting 3-manifold, as in a topological quantum field theory.

There are several other invariants for 3-manifolds Y derived from 4-manifold techniques applied to Y × R. The earliest, Instanton Floer homology, was due of course to Floer and uses the Donaldson invariants (9). Another version uses the Seiberg–Witten equations on Y × R, where the solutions on Y are called monopoles; the details appear in the monograph of Kronheimer and Mrowka (10). A third version, embedded contact homology (ECH), was created by Hutchings (11). A good survey of these theories can be found at Wikipedia.

ECH requires a contact structure on Y; it corresponds to the choice of an almost complex structure on Y × R. The contact structure on Y is given by a differential 1-form λ satisfying λ ∧ dλ > 0 everywhere (equivalently, a nowhere integrable 2-plane field on Y). A Reeb vector field ρ on Y is defined by dλ(ρ, ·) = 0 and λ(ρ) = 1; it integrates to a flow on Y that leaves λ invariant. The Reeb vector field must have closed orbits, as shown by Taubes (12) in his proof of the Weinstein Conjecture. These closed orbits, counted with multiplicity, form chain groups, and again, pseudoholomorphic curves in Y × R, which limit on these closed orbits, give a differential and then ECH.

Three of the four Floer homology theories for Y3 (not Instanton Floer homology) are expected to be essentially equivalent. Taubes (12) has proven that the Seiberg–Witten Floer homology is equivalent to ECH. In the paper of Colin et al. (13), the authors outline a proof that the hat versions of ECH and Heegaard Floer homology are equivalent (Kutluhan et al. have also announced a proof; refs. 1416). The method is to describe Y as an open book; it then has a contact structure for which the Reeb vector field is positively transverse to the pages and tangent to the binding. The Heegaard splitting is then constructed with Heegaard surface equal to the union of two pages along the binding. These tools eventually lead to the proof.

Hutchings (11) uses ECH in a different way in his paper, which addresses the question of whether one symplectic 4-manifold embeds in another. Of course, the volume of the former must not be greater than the volume of the latter. There are classic results from Gromov (17) and very recent results from McDuff (18). Gromov's nonsqueezing theorem (17) states that a symplectic 2n ball cannot be symplectically embedded in a cylinder, B2 × R2n ? 2 if the radius of the 2n ball is greater than the radius of the 2-disk. McDuff (18) considers 4-dimensional ellipsoids and gives quite subtle necessary and sufficient conditions for one ellipsoid to symplectically embed in another. Hutchings (11) extends McDuff's (18) results by using ECH capacities to give obstructions to the embedding of one symplectic 4-manifold in another (which is sharp in McDuff's cases) (18).

ECH capacities are defined for a symplectic 4-manifold X with a contact boundary Y. There is a symplectic action on the closed orbits of the Reeb vector field ρ (mentioned above) given by integrating the contact form over the orbit. The ECH capacities of X are defined in terms of the amount of symplectic action needed to represent certain classes in the ECH of Y.

Another extension of Floer homology to 3-manifolds with boundary is found in the paper of Lekili and Perutz (19). As the authors say, their version is “alarmingly abstract,” but this is the underlying topology. Given a surface Σ of genus g, the g-fold symmetric product has complex, hence symplectic, structure. A family of g disjointedly embedded circles in Σ forms a Lagrangian in the symmetric product. Two such families of circles give two Lagrangians, and they intersect generically in points. Two points may be joined by a pseudoholomorphic strip, as above, where the two edges of the strip lie in either Lagrangian, and the strip limits on the two points. If so, a differential arises, mapping one point to the other. This idea in this context goes back to Floer in the mid 1980s and accounts for his name on many theories.

Lekili and Perutz (19) work in this context and develop the special case when the boundary of Y consists of a sphere and a torus, and Y is thought of as a bordism between them.

Bourgeois et al. (20) use pseudoholomorphic curves and Floer theory in higher dimensions. The goal of their paper is computation of symplectic invariants of a large class of noncompact symplectic manifolds called Weinstein manifolds. This class includes, in particular, all complex affine varieties. A Weinstein manifold X admits a decomposition by handles, such that the core discs of handles are isotropic. In particular, indices of these handles are always ≤n, where dim X = 2n. One would like to know how symplectic invariants change when a handle is attached. The answer is known and quite simple in the case when the index of the handle is <n. Therefore, the most interesting is the case of the critical surgery, when the handle of index n is attached. In an earlier work of Bourgeois et al. (20), there was found a critical surgery formula for symplectic homology as additive groups. The current paper studies what is happening to the multiplicative structure on symplectic homology. The authors found an explicit description of the product on symplectic homology after handle-attaching in terms of certain effectively computable invariants associated with Legendrian spheres along which the index n handles are attached. In particular, for a 4-dimensional Weinstein symplectic manifold, this yields an explicit combinatorial description for its symplectic homology as a ring.

The set up is a Weinstein manifold X of dimension 2n defined by a Liouville domain X0 (a symplectic 2n-manifold made with handles of index ≤n ? 1 and with positive contact boundary ?X0) together with a collection of n handles attached to Legendrian (n ? 1) spheres in ?X0. The Legendrian spheres lie, by definition, in the (2n ? 1) planes defined by the kernel of the contact 1-form, and the Reeb vector field defined by the 1-form will have chords running from some points on one attaching sphere to points on itself or another sphere. The authors organize these chords into a graded module and define a differential using pseudoholomorphic punctured disks in ?X0 × R, which limit on one chord at one end, on multiple chords at the other end, and lie in the attaching spheres otherwise. The result is the symplectic homology of X. They then compute this homology in the simplest case, the cotangent bundle of Sn, obtained by adding one n handle to the Legendrian unknot in ?B2n.

These first five papers have made use of symplectic geometry and pseudoholomorphic curves, and this requires a symplectic 2-form that gives a nontrivial element in the second cohomology of the manifold. Donaldson's (2) original invariants only needed nontrivial second cohomology but not a symplectic structure. Attempts have been made to extend these theories to all smooth 4-manifolds, particularly to homotopy 4-spheres. The latter are homeomorphic to the standard 4-sphere, S4, but they are not known to be diffeomorphic in general. There are many possible counterexamples to this smooth 4-dimensional Poincare Conjecture, the last remaining unknown case of the general Poincare Conjecture.

It turns out that many nonsymplectic 4-manifolds have a closed 2-form ω with ω ∧ ω ≥ 0, where ω vanishes identically along a smoothly embedded 1-manifold Z ? X and is symplectic on X ? Z. One then attempts to count pseudoholomorphic curves, which limit on Z in a controlled way. Note that such closed 2-forms cannot exist on a homotopy 4-sphere. However, the notion of a 1-manifold Z, off of which ω is nice, is an idea that works well when combined with classical Lefschetz fibrations.

Recall that a Lefschetz fibration, p: X4CP1 = S2, is a complex analytic map on a complex surface X, which is locally a projection (the differential has complex rank one) or is a Lefschetz singularity, (z, w) → zw, in local coordinates. Away from the (finitely many) singularities, the map p is a bundle map with real 2-dimensional fibers that are tori in the best-understood cases. Removing a neighborhood of a torus fiber and gluing it back in a nontrivial way (a log transform) gave the first examples of exotic smooth structures on 4-manifolds (19).

The complex map p is stable over the complex numbers, but over the reals, it is not; the Lefschetz singularities perturb to a differentiable map that has fold curves, Z, with cusps. In local coordinates around a point of Z, p looks like a 1-parameter family of Morse functions, (x, y, z, t) → (±x2 ± y2 ± z2, t), or at a cusp point, (x, y, z, t) → (x3 ? tx ± y2 ± z2, t). The latter is commonly known as a birth or death of a canceling pair of critical points. These fold curves are smoothly embedded in X and play the role of the 1-manifold Z above, but they map to immersed 1-manifolds with cusps in S2.

It is important that the fibers of p: X4X2 are connected, and therefore, it is necessary (not sufficient) that the fold curves should be indefinite [that is, the signs in (x, y, z, t) → (±x2 ± y2 ± z2, t) should not all be positive, or negative]. The existence of such maps [called by different names (e.g., broken fibrations)] was first shown by Saeki (22), and a quick proof is sketched in the paper of Gay and Kirby (23). Lekili (24) introduced classical real singularity theory to the subject, giving a few natural moves that might transform one broken fibration into another for the same X. Williams (25) proved that these moves suffice for broken fibrations over a closed surface. In the paper of Gay and Kirby (23), the authors study what they call indefinite Morse 2-functions (renaming broken fibrations) and prove that the same moves suffice for maps XnB2, n ≥ 4. With this building block, they extend the theorem to all compact surfaces while keeping the fibers connected throughout the transformation from one fibration to another (23).

Williams (25) has shown that the fold curves, Z, can be chosen to be just one embedded circle Z ? X, whose projection to S2 has cusps. Thus, any smooth, closed, connected, oriented 4-manifold can be described as Fg + 1 × B2, a genus g + 1 surface cross a 2-disk lying over one component of S2 ? Z and a collection of embedded circles (not disjoint) to which 2-handles are attached when crossing an arc of Z; one then gets Fg × B2 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 immersed daisy chain in Fg + 1.

In Williams’ paper (25), he introduces four natural moves and then gets a uniqueness theorem, saying that these moves allow one to transform one simplified, purely wrinkled fibration (Williams’ name for these one-component broken fibrations) to another.

If 4-manifold invariants are defined using the combinatorics of Z in X, then the uniqueness theorems are needed to show that the invariants are invariants of the 4-manifold, not just of the particular broken fibration. One expected invariant arises from counting multisections of X4S2, which limit on Z. That is, surfaces in X whose projection back down to X2 are branched coverings of X2 with boundary Z projecting to p(Z). These multisections are similar to pseudoholomorphic curves, which also arise as multisections of Lefschetz fibrations. This extension to all 4-manifolds follows a program of Perutz (26, 27).

The study of topological smoothly embedded surfaces in 4-manifolds, with no additional geometric structure, starts with the simple case of classifying embedded, disjoint surfaces in the 4-ball, B4, each of which bounds a given knot in the 3-sphere, S3. Additionally, the simplest case of this asks for the smallest genus surface in B4, which a knot K bounds [this is the 4-ball genus as opposed to the minimal genus of a surface in S3, which is now known through the knot (Heegaard) Floer homology of Ozsvath and Szabo (28)].

There are some knots other than the unknot that bound 2-disks in B4, called slice knots, but the trefoil knot, the simplest actual knot, can be shown not to be slice by use of the most fundamental of knot invariants, the Arf invariant, whose values lie in the group of order two. Any knot in S3 bounds a Seifert surface (oriented and connected), and the trefoil knot can be drawn so as to exhibit this surface, a punctured torus, as in Fig. 1. The punctured torus is a thickened figure 8 from ref. 28, and each (darker) circle of the figure 8 from ref. 28 has a neighborhood that is a band with one full left-handed twist. In general, a knot bounds a surface, which is a boundary connected sum of g (the genus) tori. The circles of the generalized figure 8 from ref. 28 (a bouquet of circles) all lie in bands with twists. The Arf invariant of the knot is the modulo-two number of tori for which both bands have an odd number of twists.

Fig. 1.

Trefoil knot.

The Arf invariant has higher-order generalizations as do the linking numbers of the components of a link. Conant et al. (29) use these generalizations to give a nearly complete answer to the problem of classifying the Whitney towers that a link can bound in the 4-ball. A link may not bound disjoint surfaces, and therefore, the authors immerse 2-disks, each of which bounds a component of the link, and then try to remove pairs of intersections using Whitney disks. These may be immersed, and so again, they try to remove intersections with Whitney disks and so on. The result is Whitney towers, which seem to be the most natural tool for studying these questions.

Just like Milnor invariants are higher-order generalizations of the linking numbers between two components of a link, Conant et al. (29) propose order n generalizations of Arf invariants for links (with values in certain Z2 vector spaces Vn). The Arf invariant for knots explained above is the case V1 = Z2, and modulo is the determination of the higher-order spaces Vn (only upper bounds are given); the paper classifies Whitney towers that a link can bound in the 4-ball. A link may not be slice (i.e., not bound disjoint embedded disks but immersed disks always exist in the 4-ball). The authors try to remove pairs of intersections between such disks using Whitney disks. These may be immersed, and therefore, they try to remove intersections with Whitney disks and so on. If n layers of such Whitney disks are chosen, the resulting 2-complex is called a Whitney tower of order n. It has an intersection invariant, reflecting all order n Milnor and Arf invariants of the link on the boundary of the Whitney tower. The obstruction theory in the paper is based on the result that this intersection invariant vanishes if and only if the construction can be continued one more step [i.e., the given link bounds, in fact, a Whitney tower of order (n + 1)].

The picture on the cover page of this volume is related to the main new maneuver on Whitney towers. It starts with four small spheres in 4-space that are disjoint; one of them is shown locally as the horizontal plane in the picture. Then, each of the other 3-spheres is made to intersect this plane, each in two points and each paired by a Whitney disk. After dragging around the boundaries of the Whitney disks in the horizontal plane (the bounding circles are drawn in red, blue, and black in the picture), they form the Borromean rings. It turns out that the intersection invariant of the resulting Whitney tower has exactly three terms that correspond to the terms in the Jacobi relation for a Lie algebra. Since the original spheres were disjoint, this relation must also be introduced into the obstruction theory for Whitney towers and it is the key in translating the intersection invariant into Milnor and Arf invariants.

Morrison and Walker (30) introduce a rather abstract notion called the blob complex. To motivate their definition, one can start with the familiar fundamental group of a topological space X with base point x0 and describe five independent directions in which the fundamental group construction can be generalized. The blob complex can be thought of as combining all five of these generalizations: whereas π1(X) captures maps from the interval into X, the blob complex B*(M ; C) provides a notion of maps from any manifold M into an n category C.

Recall that the fundamental group π1(X, x0) is given by homotopy classes of loops in X based at x0, thought of as maps of the interval [0, 1] into X that send the endpoints to x0. Two loops, α and β, can be multiplied by first traversing α and then β. This multiplication is associative but only up to homotopy. (In this case, the images of the loops coincide, but they are parametrized differently.)

In the first generalization of the fundamental group, homotopy classes of loops are replaced with actual loops. Loops (based at x0) can still be multiplied, but this multiplication will no longer be associative. It will, however, be associative up to deformations, and these deformations can be incorporated into a more complicated structure. It turns out to be useful to consider deformations between deformations and so on ad infinitum. The resulting structure incorporating loops, multiplication and iterated deformations is called an A space [whose connected components happen to form a group, namely π1(X, x0)].

In the second generalization, paths are considered instead of loops, the endpoints of the interval [0, 1] can map to different points of X. Paths can be multiplied but only if the final endpoint of the first path coincides with the initial endpoint of the second path. The resulting algebraic structure is a category (with invertible morphisms) rather than a group.

In the third generalization, loops are replaced with maps of n-dimensional spheres into X. This old notion, πn(X, x0), goes back to the 1930s. It can be combined with the second generalization to consider n-dimensional paths (that is, maps of the n ball Bn into X). These maps can be multiplied (i.e., glued together) in various ways when their boundaries partially coincide. The resulting algebraic structure is a standard example of an n-category. Adding in the first generalization (distinguishing deformation equivalent maps) gives an example of an A n-category. One can think of an n-category as an algebraic structure with properties that mimic the ways in which maps from Bn to X can be glued together.

In the fourth generalization, n-balls are replaced with arbitrary n-manifolds Mn and maps of Mn into X. Since Mn can be carved up into copies of the standard ball Bn, these mapping spaces are in some sense determined by the maps from Bn to X of the previous paragraph. Put another way, maps from Bn to X are a local invariant that can be integrated to construct the global invariant of maps from Mn to X.

In the fifth and final generalization, note that the integration procedure of the previous paragraph works just as well if the maps from Bn to X are replaced by maps to any n-category or A n-category. This is essentially what the blob complex is: a way of combining an n-manifold Mn and an A n-category C to produce a chain complex B*(Mn; C), which can be thought of as the chains on the space of maps from M to C.

Lipshitz et al. (8) show that the Heegaard–Floer spaces of 3-manifolds satisfy a gluing law that involves taking a tensor product over a certain A algebra associated to a 2-manifold. The blob complex also has an A tensor product gluing law like this, and therefore, it is natural to ask if there is some A three-category HF such that HF*(M3) ? HF*(M3;HF).

Another possible application for the blob complex to 4-manifolds concerns Khovanov homology. Modulo some sign issues, one should be able to construct a 4-category K from Khovanov homology and then associate the chain complex B*(W4; K) to a 4-manifold W.


This work was supported in part by US National Science Foundation Grant EMSW21-RTG.


  • ?1E-mail: kirby{at}math.berkeley.edu.
  • Author contributions: R.C.K. wrote the paper.

  • The author declares no conflict of interest.


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