Poincare's conjecture and the homeotopy group of a closed, orientable 2-manifold

Abstract

In 1904 Poincaré [11] conjectured that every compact, simply-connected closed 3-dimensional manifold is homeomorphic to a 3-sphere. The corresponding result for dimension 2 is classical; for dimension ≧ 5 it was proved by Smale [12] and Stallings [13], but for dimensions 3 and 4 the question remains open. It has been discovered in recent years that the 3-dimensional Poincaré conjecture could be reformulated in purely algebraic terms [6, 10, 14, 15] however the algebraic problems which are posed in the references cited above have not, to date, proved tractable.