I started reading a Morse theory by Milnor and am not understanding something.(adsbygoogle = window.adsbygoogle || []).push({});

I am reading the proof of Theorem on page 25:

Let M be a compact manifold and f be a differentiable function on M with only two critical points, both of which are non-degenerate, then M is homeomorphic to a sphere.

We may assume that 0 is mimimum and 1 is maximum of f.

In the proof he says that by Morse lemma for small epsilon f^-1[0, epsilon], f^-1[1-epsilon, 1] are closed n-cell.

I guess he is using the fact that on f^-1(1) and f^-1(0) the morse index is 0.

But Is the fact obvious?

I know 0 and 1 are minimum and maximum respectively. So Hessian is positive at f^-1(0) and negative at f^-1(1).

But why is morse index 0 at these points?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Morse Index

**Physics Forums | Science Articles, Homework Help, Discussion**