User stephan wiesendorf - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T12:01:05Z http://mathoverflow.net/feeds/user/12137 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75408/perfect-morse-functions Perfect Morse functions Stephan Wiesendorf 2011-09-14T14:42:27Z 2011-09-15T23:16:48Z <p>Does anyone know a concrete example of a Morse function on some manifold that is perfect with respect to some field but not with respect to $\mathbb Z_2$?</p> 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/75408/perfect-morse-functions/75557#75557 Comment by Stephan Wiesendorf Stephan Wiesendorf 2011-09-17T07:46:30Z 2011-09-17T07:46:30Z I had suspected that Lens spaces might provide an example, but I was not familiar with their description via the Heegaard splitting. When I thought about your first comment, it was not clear to me what is meant by a standard Morse function coming from such a splitting, but I assumed that it should be obtained on each solid torus from a perfect Morse function on the core circle extended in such a way that it is constant on the boundary. I was pretty sure that this can be done and if I understand your expansion right this is exactly what you describe (reversed). So, thank you again. http://mathoverflow.net/questions/75408/perfect-morse-functions Comment by Stephan Wiesendorf Stephan Wiesendorf 2011-09-15T09:17:36Z 2011-09-15T09:17:36Z Ok, I see. Thank you. http://mathoverflow.net/questions/51661/restrictions-of-perfect-morse-functions-to-submanifolds/51666#51666 Comment by Stephan Wiesendorf Stephan Wiesendorf 2011-01-10T20:54:57Z 2011-01-10T20:54:57Z Ok, I see. Thank you. The problem is that the attaching map restricts to a map of degree 2. Is that a problem of orientability? Can one avoid problems like that by using $\mathbb Z/2\mathbb Z$-coefficients, for instance?