# Poincaré Conjecture (1900)

### From Mathsreach

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*