5

The only closed manifolds which allow a function with two (maybe degenerate) critical points are spheres. In dimension 2 it is quite easy to prove, but what is about higher dimensions?

Community
  • 1
evgeny
  • 3,781
  • Given Jim's answer below, were you looking for a proof in the topological case, where there is a positive result? That is, every compact smooth manifold which admits exactly two non-degenerate critical points is homeomorphic (not necessarily diffeomorphic) to a sphere? – Dan Rust Oct 13 '14 at 12:57
  • Morse Theory by John Milnor has a full proof of the topological result mentioned by Grumpy Parsnip. – Tim kinsella Oct 13 '14 at 13:04
  • 4
    @Timkinsella Milnor mentions that the possibly degenerate case is more difficult than the proof given in Morse Theory, but he gives two references for proofs of this more general case. – Dan Rust Oct 13 '14 at 13:07
  • @DanielRust Ah, thanks. I should have checked. – Tim kinsella Oct 13 '14 at 13:12
  • The proof for the non-degernate case is too simple to not mention though. A good example of the power of the Morse lemma. – Dan Rust Oct 13 '14 at 13:14
  • @Daniel Rust: Yes, I meant it. Wikipedia link by Grumpy Parnis gives [this article][http://www.maths.ed.ac.uk/~aar/papers/exotic.pdf], where it is quite easily proven if one of points is non-degenerate. – evgeny Oct 13 '14 at 13:15

1 Answers1

6

Topologically the answer is yes, but in the smooth category the answer is no. Milnor provided the first examples of topological spheres which are not diffeomorphic to the standard sphere. His argument that they are toplogical spheres uses the fact that there are Morse functions with two critical points. This wikipedia page should get you started .

Cheerful Parsnip
  • 27,278