How to Solve the Poincare Conjecture

The Poincare Conjecture, initially posed as a topographical question, was solved by Grigori Perelman after nearly a century of work by mathematicians trying to solve the question. The Poincare Conjecture addresses the nature of spheres. Because the conjecture has been solved, it is now considered a theorem.

Instructions

1. Define the difference between a 3-sphere and a 3-manifold, as originally posed by Henri Poincare. Poincare's question was whether a 3-manifold with a trivial fundamental group was necessarily a 3-sphere.

2. Define "trivial fundamental group" as the quality of a sphere where every loop drawn on the surface can be shrunk to a single point.

3. Understand Poincare's original phrasing, which asked if a compact 3-dimensional manifold without boundary could have a fundamental group that was trivial (like a 3-sphere) if the manifold was not a 3-sphere.

4. Rephrase the original statement as the standard conjecture, which is that every simply connected, compact 3-manifold (without boundary) is homeomorphic to the 3-sphere.

5. Solve the n=1 and n=2 case of the conjecture by knowing that the n=1 case is known to be trivial and the n=2 case is known to be classical. The n=3 case is the original conjecture, proved by Perelman.

6. Use Thurston's Geometrization Conjecture to find that Perelman's solution to the Poincare Conjecture follows from the result. Thurston's Conjecture showed that, after splitting a 3-manifold into its connected sum and the Jaco-Shalen-Johannson torus decomposition, the result is that remaining components each fits exactly one of 8 specified geometries. Poincare's problem is a subset of Thurston's.

Tips & Warnings

• Understand the Poincare Conjecture is a question of topography, so that if you are well versed in algebra and combinatorics, you can apply those principles to topography to understand the nature of Poincare's question.