Restrictions of perfect Morse functions to submanifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:43:25Z http://mathoverflow.net/feeds/question/51661 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51661/restrictions-of-perfect-morse-functions-to-submanifolds Restrictions of perfect Morse functions to submanifolds Stephan Wiesendorf 2011-01-10T15:20:14Z 2011-01-10T15:45:22Z <p>A Morse function $f: M \rightarrow \mathbb R$ on a connected closed manifold $M$ is called $\mathit{perfect}$ with respect to the field $\mathbb F$ if all of the Morse inequalities are equalities, i.e. the number of critical points of $f$ with index $k$ coincides with the $k$-th Betti number of $M$ with respect to $\mathbb F$-coefficients for all $k$.</p> <p>Now assume that $f: M \rightarrow \mathbb R$ is a perfect Morse function on a closed connected Riemannian manifold and that $N \subset M$ is a closed submanifold, such that $N$ contains all the critical points of $f$ and the restriction $f|_N$ is a Morse function on $N$. If in addition the gradient $\nabla f$ is tangent to $N$ along $N$, i.e. $\nabla f|_N \in \Gamma(TN)$, then $N$ is a union of flow lines and the critical points of $f|_N$ are exactly the critical points of $f$. </p> <p>Now my question is: Is it possible to deduce perfectness of $f|_N$ from perfectness of $f$ in this setting, or are there additional conditions under which this is possible? </p> http://mathoverflow.net/questions/51661/restrictions-of-perfect-morse-functions-to-submanifolds/51666#51666 Answer by Tom Goodwillie for Restrictions of perfect Morse functions to submanifolds Tom Goodwillie 2011-01-10T15:45:22Z 2011-01-10T15:45:22Z <p>Isn't there a counterexample with $(M,N)=(\mathbb CP^2,\mathbb RP^2)$?</p>