Geometrization and the Poincaré conjecture

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1 Geometrization and the Poincaré conjecture Jan Metzger BRIGFOS, 2008

2 History of the Poincaré conjecture In 1904 Poincaré formulated his conjecture. It is a statement about three dimensional geometric objects, but has been generalized to other dimensions as well. In two dimensions its solution was known to Poincaré. In 1960 Smale presented a proof for dimension n > 4 and got the Fields medal. In 1982 Friedman solved dimension 4 and got the Fields medal. In 2000 the Clay Institute named the Poincaré conjecture one of the seven Millennium Problems and offered 1 Million US-$ for its solution. In 2002 Perelman published his breakthrough papers which lead to the solution of the Poincaré conjecture via Hamilton s Ricci-flow approach.

3 Surfaces Definition A two dimensional manifold or surface is a geometric object, which can be covered by charts. A chart is a patch of the plane. Remark The charts make a surface locally look like a plane. The global structure can be very different from the plane. Examples Plane Sphere Torus

4 Geometry on surfaces: The torus Geometry Geometry is about measuring lengths and angles. Example: The Torus Lengths and shortest connections Angles and sum of angles Sum of angles in triangle is as in the plane The torus has a flat geometry

5 Geometry on the sphere Example: The sphere Lengths and shortest connections Angles and sum of angles Sum of angles is larger than in flat space Angle defect is positive The sphere has positive curvature

6 Geometry on the hyperbolic plane Example: The hyperbolic plane Covered by one chart, the unit disk Angles as in flat space Lengths are stretched by the factor P 3 P 1 P r 2 r is the distance to the origin Shortest connections are circles Angle defect is negative Hyperbolic plane has negative curvature

7 Three dimensions Definition A three dimensional manifold or 3-manifold is a geometric object, which can be covered by charts. Here a chart is a patch of 3-space. Examples 3-Torus 3-Sphere Hyperbolic 3-space 3-sphere The 2-sphere is a union of circles which close up. Similarly the 3-sphere is a union of 2-spheres which close up.

8 Topology: Rubber band geometry Concept We do not distinguish manifolds that can be continuously be deformed into each other. Example We do not distinguish the surface of donut or a cup.

9 Classification of manifolds Question Can we describe all the different possibilities for deformation types? Idea Try to find standard representatives for each deformation type. Standard representatives should be distinguished by particularly nice geometric properties, like high symmetry. This question has two different aspects: 1. Find and categorize all standard representatives. 2. Deform any arbitrary manifold to one of the standard representatives. The second step is particularly difficult, as we need an algorithm to decide to which standard representative a given manifold can be deformed (otherwise we would be done).

10 The classification of surfaces For surfaces the situation is completely known An oriented surface is either a sphere, or a surface that arises from glueing g copies of the torus. Remark One can decide in advance of which type a surface is by counting holes. Each topological type has a representative with constant curvature. We say that each surface is geometrizable.

11 Geometrization Question Can every 3-manifold be geometrized? Can we deform every three dimensional manifold to a standard model? Observation The situation in three dimensions is more complicated: There are more than three standard geometries, in fact there are eight. Not every deformation type corresponds to one of these geometries. Geometrization Any 3-manifold can be cut into pieces such that each piece is geometrizable. The decomposition is entirely determined by topological information of the deformation type.

12 The Poincare conjecture Statement Every 3-manifold with the property that each closed curve can be contracted to a point can be deformed to the 3-sphere. Two dimensions In two dimensions, this follows from the classification. Whenever a surface has a hole, there is a curve which can not be contracted. The only type with no holes is the sphere.

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