Index: number of linearly independent direction you can move for which the function decreases.
index of a Morse function
-
Morse lemma
Nondegenerate critical points have standard forms \(\sum \pm x_i^2\), so the index of a Morse function is well-defined.
-
Morse Theory
Definition: index of a Morse function
Theorem: \(M\) is homotopy equivalent to a CW complex with one cell of dimension \(k\) for each critical point of \(f\) of index of a Morse function \(k\).