Question

As well known, Perelman proved Poincare conjecture by proving Thurston's Geometrization conjecture.

Somebody says that we can understand part of the universe from Poincare conjecture.

As a purely topological viewpoint, why do you think the poincare conjecutre is important and how about Smooth poincare conjecture in dimension 4?

One may simply answer that because it is extremely difficult to prove or because it made 100 years of development of history of Topology or geometry....

But I want to know your professional, new, own viewpoint.

Answer 1

Mine is no profesional, and certainly don't believe its new nor own, but I'll give it a try.

In my opinion, algebraic topology tries to caracterize nice topological spaces (say CW complexes) modulo homotopy equivalence (which is the reasonable equivalence given the fact that the invariants used are usually invariants under homotopy equivalence). This caracterization has a good important theorem (for me) which is Whitehead's theorem.

When studying manifold topology, one would like to get clasification modulo homeomorphisms so, the above study it is not enough. This gives great importance to theorems such as the clasification of spheres by homotopy type.

I think it is interesting that the result from this point of view had been solved for tori (which are much simpler from the point of view of higher homotopy groups) by Hsiang and Wall (and maybe others).

In brief, I believe that Poincare Conjecture it is one of the central and most natural questions one can pose in manifold topology (or geometric topology?).