V.25 The Poincaré Conjecture

The Poincaré conjecture is the statement that a COMPACT [III.9] smooth n-dimensional manifold that is HOMOTOPY EQUIVALENT [IV.6 2] to the n-sphere Sⁿ must in fact be homeomorphic to Sⁿ. One can think of a compact manifold as a manifold that lives in a finite region of ^m for some m and that has no boundary: for example, the 2-sphere and the torus are compact manifolds living in ³, while the open unit disk or an infinitely long cylinder is not. (The open unit disk does not have a boundary in an intrinsic sense, but its realization as the set {(x,y) : x² + y² < 1} has the set {(x, y) : x² + y² = 1} as its boundary.) A manifold is called simply connected if every loop in the manifold can be continuously contracted to a point. For instance, a sphere of dimension greater than 1 is simply connected but a torus is not (since a loop that “goes around” the torus will always go around the torus, however you continuously deform it). In three dimensions, the Poincaré conjecture asks whether two simple properties of spheres, compactness and simple connectedness, are enough to characterize spheres.

The case n = 1 is not interesting: the real line is not compact and a circle is not simply connected, so the hypotheses of the problem cannot be satisfied. POINCARÉ [VI.61] himself solved the problem for n = 2 early in the twentieth century, by completely classifying all compact 2-manifolds and noting that in his list of all possible such manifolds only the sphere was simply connected. For a time he believed that he had solved the three-dimensional case as well, but then discovered a counterexample to one of the main assertions of his proof. In 1961, Stephen Smale proved the conjecture for n ≥ 5, and Michael Freedman proved the n = 4 case in 1982. That left just the three-dimensional problem open.

Also in 1982, William Thurston put forward his famous geometrization conjecture, which was a proposed classification of three-dimensional manifolds. The conjecture asserted that every compact 3-manifold can be cut up into submanifolds that can be given METRICS [III.56] that turn them into one of eight particularly symmetrical geometric structures. Three of these structures are the three-dimensional versions of Euclidean, spherical, and hyperbolic geometry (see [I.3 §6]). Another is the infinite “cylinder” S² × : that is, the product of a 2-sphere with an infinite line. Similarly, one can take the product of the hyperbolic plane with an infinite line and obtain a fifth structure. The other three are slightly more complicated to describe. Thurston also gave significant evidence for his conjecture by proving it in the case of so-called Haken manifolds.

The geometrization conjecture implies the Poincaré conjecture; both were proved by Grigori Perelman, who completed a program that had been set out by Richard Hamilton. The main idea of this program was to solve the problems by analyzing RICCI FLOW [III.78]. The solution was announced in 2003 and checked carefully by several experts over the next few years. For more details, see DIFFERENTIAL TOPOLOGY [IV.7].