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

Published online by Cambridge University Press:  09 April 2009

Joan S. Birman
Joan S. Birman
Affiliation:
Stevens Institute of Technology, Castle Point StationHoboken, N. J. 07030, U. S. A.
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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.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 17 , Issue 2 , March 1974 , pp. 214 - 221
Copyright
Copyright © Australian Mathematical Society 1974

References

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