The Poincaré Conjecture (A bit offtopic entertainment) Coffetalk, Alexander Mey 05.04.2016

Every simply connected, closed 3-manifold is homeomorphich to 3-Sphere Goal: Understand the Poincaré Conjecture: Every simply connected, closed 3-manifold is homeomorphich to 3-Sphere

History Henri Poincaré (1854-1912) was interested in what topological properties fully describe a sphere

History Henri Poincaré (1854-1912) was interested in what topological properties fully describe a sphere First he thought a concept called homology could fully describe it, but he constructed the so called homology sphere as a counterexample (a 3 manifold)

History Henri Poincaré (1854-1912) was interested in what topological properties fully describe a sphere First he thought a concept called homology could fully describe it, but he constructed the so called homology sphere as a counterexample (a 3 manifold) He introduced a new topological invariant, the so called fundamental group, that could show that a homology sphere and a 3-sphere are indeed (topologically) different

History Henri Poincaré (1854-1912) was interested in what topological properties fully describe a sphere First he thought a concept called homology could fully describe it, but he constructed the so called homology sphere as a counterexample (a 3 manifold) He introduced a new topological invariant, the so called fundamental group, that could show that a homology sphere and a 3-sphere are indeed (topologically) different Question arised: What if we assume that a 3-manifold has trivial fundamental group (simply connected), do we then know that it is a 3-sphere?

Basic concepts of topology A topological space is a set X together with a set of subsets of X that define the topology. The set of subsets are called the open sets Two topological spaces are seen to be topologically the same if they are homeomorphic Intuition about hoemeomorphism: You can transform one space into the other by stretching, contracting and bending but without cutting and gluing

A donut is the same as a cup

Fundamental group The fundamental group counts the amount of different (up to topological transformation) loops in a space Simply connected:

Now we understand: Every simply connected, closed 3-manifold is homeomorphich to 3-Sphere

Some more history Grigori Perelman presented a proof in three papers made available in 2002 and 2003 on arXiv The work was confirmed in 2006, and he was awarded the Fields Medal, which he declined In 2010 he was awarded the Millennium Prize which he also declined saying that his work was not greater than Hamilton’s (1943), a mathematician who introduced an important tool for the proof A similar statement for higher dimensions was thought to be false, but proven first in 1961

Thanks