Low-dimensional topology and geometry.

A t the core of low-dimensional topology has been the classification of knots and links in the 3-sphere and the classification of 3and 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 X. 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 4space, R. 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 X, the Seiberg–Witten invariants were equivalent to the Gromov–Witten invariants, which count the number of pseudoholomorphic curves in X that belong to certain 2dimensional 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 toM × 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 Y (6, 7). This can be derived from a Morse function f: Y → R and the index zero and one critical points of f have a neighborhood (the 0and 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://en.wikipedia.org/wiki/ Grigori_Perelman) using differential geometric methods, not a better understanding of the mapping class group.) The homology of Y 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 1dimensional 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 2plane 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 Y (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. 14–16). 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 4manifold embeds in another. Of course, the volume of the former must not be greater than the volume of the latter.