What's the deal with Poincare?

This is a rather interesting article ...but I don't get what's the big deal. The article says that it is not only famous, but important.

Important how? I mean, I see how it could be used to make some quirky morphing animations...but that's not really important, and according to the article they already had the algorithm to do it, now they just want to "prove" it.

I can't imagine how knowing that any closed surface in an arbitrary dimension can be mapped as the same way as another surface with the same number of holes can really be used to say anything important about spacetime or string theory, but I assume there is some obscure unconving relationship that this can be used to cobble together...but please do explain, if you know it!