Poincaré Conjecture (1900)

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If M is a 3-manifold with trivial fundamental group, and Π_i(M)=0 for
i=1,2 and =Z for i=0,3 (ie, M has the homotopy groups of a 3-sphere), then
M is homeomorphic to the 3-sphere.

Simply: (1904) That if any loop on the surface of a three-dimensional shape can
be shrunk to a point (as any loop can on a 3-D sphere) then the shape is just a 3-D

Discipline: Topology

Originator: Jules Henri Poincaré, 1854-1912.

Incentive: $US1million, one of the seven Millennium Prize Problems of the Clay
Mathematics Institute.

Notable false proof: JHC Whitehead, 1934.

Has led to: Interesting new examples of 3-manifolds; several celebrated cases of
Poincaritis.

Unusual aspect: Solving this problem in four and more dimensions has been much
easier than solving it in three.

Likely proof: Grigori Perelman, Steklov Institute of Mathematics, St Petersberg,
2002 and 2003, although the Clay prize has yet to be awarded.

NZIMA programme connection:
Geometric Methods in the Topology of 3-Dimensional Manifolds.

Published in IMAges 1 - October 2006