The Morse-Bott inequalities via dynamical systems - Mathematics > Algebraic TopologyReport as inadecuate




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Abstract: Let $f:M \to \mathbb{R}$ be a Morse-Bott function on a compact smooth finitedimensional manifold $M$. The polynomial Morse inequalities and an explicitperturbation of $f$ defined using Morse functions $f j$ on the criticalsubmanifolds $C j$ of $f$ show immediately that $MB tf = P tM + 1+tRt$,where $MB tf$ is the Morse-Bott polynomial of $f$ and $P tM$ is thePoincar\-e polynomial of $M$. We prove that $Rt$ is a polynomial withnonnegative integer coefficients by showing that the number of gradient flowlines of the perturbation of $f$ between two critical points $p,q \in C j$coincides with the number of gradient flow lines between $p$ and $q$ of theMorse function $f j$. This leads to a relationship between the kernels of theMorse-Smale-Witten boundary operators associated to the Morse functions $f j$and the perturbation of $f$. This method works when $M$ and all the criticalsubmanifolds are oriented or when $\mathbb{Z} 2$ coefficients are used.



Author: Augustin Banyaga, David Hurtubise

Source: https://arxiv.org/







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