Are there big implications of Poincare conjecture?

Question

I was just curious: are there any big corollaries of Poincare conjecture in dimension $3$? Is it useful to prove some other (big) theorems? Or is it just a nice statement, and its main value is that whyle trying to prove it, people developed a lot of machinery and important theory?

Answer 1

There's been a lot of work done on classifying manifolds up to homeomorphism, diffeomorphism, homotopy equivalence, etc. The basic question is whether a homotopy sphere in dimension $n$ is actually "equivalent" to $S^n$. I put "equivalent" in quotes because there are many different categories in which you ask the question: smooth, topological, PL, etc. More or less, dimensions $1$ and $2$ are trivial; dimension $4$ is weird (see, for example, Freedman's theorem and Rokhlin's theorem for examples of what the results look like, and look at the status of the smooth Poincare conjecture in dimension $4$); and the case of dimension $\geq 5$ is accessible by surgery methods. (I'm glossing over a huge amount here, so take the previous sentence with a healthy amount of salt.) Dimension $3$ is more geometric than algebraic, though, and required special treatment. The point is that the tools that went into proving the Poincare conjecture are themselves very significant. More generally, Perelman proved the Poincare conjecture as part of Thurston's geometrization conjecture, which gives a nice and useful classification of closed $3$-manifolds.