3-manifold

Summary

A $3$-manifold is a topological manifold of dimension $3$: a space that looks like ordinary space $\R^3$ near each of its points. By Moise's theorem every $3$-manifold has an essentially unique triangulation. It also has an essentially unique smooth structure, so the topological, piecewise linear and smooth theories agree. Their study combines topology with geometry, through the geometrization conjecture, proved by Perelman, which includes the Poincaré conjecture.