Please what is the basic idea for finding critical point via Morse theory and critical groups?
Thank you
2026-03-25 16:06:58.1774454818
Basic idea for finding critical point via Morse theory
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@Vrouvrou: You have asked many questions on the subject of Morse theory. I have the feeling any book on the subject must discuss this on the first page.
The basic observation is (under suitable compactness assumptions) the following. Let $f:M\rightarrow \mathbb{R}$ be a smooth function. And $c-\epsilon$ and $c+\epsilon$ are regular values such that the homology of the sublevel sets $f_{c-\epsilon}$ and $f_{c+\epsilon}$ are different, then $f$ posses a critical value between $c-\epsilon$ and $c+\epsilon$. This allows one to find critical points without locating them precisely from very coarse data.
Suppose a critical point is isolated and is the unique critical point with that given critical value. Then the critical group of the critical point measures how the homology changes when passing through the critical value.