Poincaré ConjectureIn its original form, the Poincaré conjecture states that every simply connected closed three-manifold is homeomorphic to the three-sphere (in a topologist's sense) , where a three-sphere is simply a generalization of the usual sphere to one dimension higher. More colloquially, the conjecture says that the three-sphere is the only type of bounded three-dimensional space possible that contains no holes. This conjecture was first proposed in 1904 by H. Poincaré (Poincaré 1953, pp. 486 and 498), and subsequently generalized to the conjecture that every compact -manifold is homotopy-equivalent to the -sphere iff it is homeomorphic to the -sphere. The generalized statement reduces to the original conjecture for .
The Poincaré conjecture has proved a thorny problem ever since it was first proposed, and its study has led not only to many false proofs, but also to a deepening in the understanding of the topology of manifolds (Milnor). One of the first incorrect proofs was due to Poincaré himself (1953, p. 370), stated four years prior to formulation of his conjecture, and to which Poincaré subsequently found a counterexample. In 1934, Whitehead (1962, pp. 21-50) proposed another incorrect proof, then discovered a counterexample (the Whitehead link) to his own theorem.
The case of the generalized conjecture is trivial, the case is classical (and was known to 19th century mathematicians), (the original conjecture) appears to have been proved by recent work by G. Perelman (although the proof has not yet been fully verified), was proved by Freedman (1982) (for which he was awarded the 1986 Fields medal), was demonstrated by Zeeman (1961), was established by Stallings (1962), and was shown by Smale in 1961 (although Smale subsequently extended his proof to include all ).
The Clay Mathematics Institute included the conjecture on its list of $1 million prize problems. In April 2002, M. J. Dunwoody produced a five-page paper that purports to prove the conjecture. However, Dunwoody's manuscript was quickly found to be fundamentally flawed (Weisstein 2002). A much more promising result has been reported by Perelman (2002, 2003; Robinson 2003). Perelman's work appears to establish a more general result known as the Thurston's geometrization conjecture, from which the Poincaré conjecture immediately follows (Weisstein 2003). Mathematicians familiar with Perelman's work describe it as well thought-out and expect that it will be difficult to locate any substantial mistakes (Robinson 2003, Collins 2004). In fact, Collins (2004) goes so far as to state, "everyone expects [that] Perelman's proof is correct."