MathWorld Headline News

Poincaré Conjecture Purported Proof Perforated

By Eric W. Weisstein

April 18, 2002--A famous unproven conjecture in mathematics states that every simply connected closed 3-manifold is homeomorphic to the 3-sphere. This conjecture was first proposed in 1904 by H. Poincaré (Poincaré 1953, pp. 486 and 498), it was subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere if and only if it is homeomorphic to the n-sphere. The generalized statement is known as the Poincaré conjecture and it reduces to the original conjecture for n = 3.

The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical, n = 3 remains open, n = 4 was proved by Freedman in 1982 (for which he was awarded the 1986 Fields medal), n = 5 by Zeeman in 1961, n = 6 by Stallings in 1962, and n >= 7 by Smale in 1961. (Smale subsequently extended his proof to include n >= 5.)

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, according to the rules of the Clay Institute, the paper must survive two years of academic scrutiny before the prize can be collected. It is unclear as of this writing if Dunwoody's proof will last even a fraction of that duration.

In fact, it appears that the purported proof has already been found lacking, judging by the facts that (1) the abstract begins, "We give a prospective [italics added] proof of the Poincaré Conjecture" and (2) the revised April 11 version of the preprint contains a small but significant change in title from "A Proof of the Poincaré Conjecture" to "A Proof of the Poincaré Conjecture?"" In particular, a critical step in the paper appears to remain unproven, and Dunwoody himself does not see how to fill in the missing proof.

Postscript: see "Poincaré Conjecture Proved--This Time for Real" for more recent results