Is it true that $RP^n$ has a Morse function with n critical points,and dont have Morse function with n-1 critical points
2026-03-27 15:16:46.1774624606
Morse functions with minimal number critical points
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Your first question: no. A Morse function on $RP^1 \cong S^1$ has at least two critical points (a maximum and minimum).
More generally, the Morse inequalities tell us that the number of critical points of index $k$ of a Morse function is bounded below by the rank of the $k$th homology group of the space in any field. For $RP^n$, we look at $H_k(RP^n, \mathbb{Z}/2)$, $0 \leq k \leq n$, and see that a Morse function on $RP^n$ must have at least $n+1$ critical points.