4
$\begingroup$

Wolfram states "The n=1 case of the generalized conjecture is trivial, the n=2 case is classical (and was known to 19th century mathematicians)"

How is it proved that every simply connected closed two-manifold is homeomorphic to the two-sphere?

And the n>3 case is known since 1962, what makes the n=3 case so hard?

  • 0
    I guess it should say 'compact' rather than 'closed' in the question, since otherwise the plane (a 2-manifold that is both simply connected and closed) would be a counter example?2017-11-28

1 Answers 1

5

Poincare for $n=2$ is contained in the classification theorem for surfaces (and that phrase should get you started, if you want to search for a proof), which says that every compact surface is homeomorphic to a sphere with some number of handles or cross-caps attached.

I once heard an expert "explain" the difficulty of the $n=3$ case to a general audience by saying something like this: when $n\le2$, there isn't enough room for anything to go wrong, while for $n\ge4$, there's enough room to fix anything that goes wrong; for $n=3$, there's enough room for something to go wrong, and (this was 15 years ago) it's not clear whether there's enough room to fix things when they go wrong.

  • 1
    "R. H. Bing explained the dimension situation in this way: “Dimension 4 is the most difficult dimension. It is too old to spank, the way we might deal with the little dimensions 1, 2, and 3; but it is also too young to reason with, the way we deal with the grown-up dimensions 5 and higher.”" Quoted from [James W. Cannon's review of *Embeddings in Manifolds*](http://www.ams.org/journals/bull/2011-48-03/S0273-0979-2011-01320-9/S0273-0979-2011-01320-9.pdf)2012-03-20