Sunday, 13 October 2013

open problem - The importance of Poincare Conjecture or SPC4?

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?).

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