Previously, we defined the mathematical structure known as a manifold which is a topological space that is locally homeomorphic to R^n (and hence able to have the notion of a coordinate system or systems).
You may recall that when we started our journey, we began with the idea of adding structure to sets and took a step down the path of topology by introducing the notion of open sets which allowed us to, in turn, define the notion of continuity. That path led us to manifolds. If we continue down the path we'll get into analysis.
But at this point, we're going to go back to sets and take a different path; rather than take the path of continuity we'll take the path of discreteness. Where as topology took us from sets to topological spaces to manifolds and the gateway to analysis, we will now explore the beginnings of group theory which will take us from sets to groups and the beginnings of abstract algebra.
Once we've spent a little time on group theory, we'll be ready to talk about the Poincaré Conjecture itself and also start laying the foundation for differential geometry, which is the basis for recent work on the conjecture as well as for Einstein's General Theory of Relativity.
UPDATE: next post
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.