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Topology/Geometry Seminar
Thursday, October 11, 2007
5:00pm,
106 McAllister
David Hurtubise
Penn State
The Morse-Bott inequalities
Let $f:M \rightarrow \mathbb{R}$ be a Morse-Bott function on a compact
finite dimensional manifold $M$. The polynomial Morse inequalities and an
explicit perturbation of $f$ defined using Morse functions on the critical
submanifolds of $f$ show immediately that $MB_t(f) = P_t(M) + (1+t)R(t)$,
where $MB_t(f)$ is the Morse-Bott polynomial of $f$ and $P_t(M)$ is the
Poincar\'e polynomial of $M$. We prove that $R(t)$ is a polynomial with
nonnegative integer coefficients by studying the kernels of the
Morse-Smale-Witten boundary operators associated to the Morse functions on
the critical submanifolds of $f$. Our method works when $M$ and all the
critical submanifolds are oriented or when $\mathbb{Z}_2$ coefficients are
used.
We meet weekly on Thursday afternoons, with the third meeting of each month in Altoona. If you are interested in giving a talk in our seminar please contact one of the coordinators for the Spring 2006 semester: Aissa Wade and Wojciech Dorabiala.
