Perfect discrete morse functions on connected sums

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2017
Varlı, Hanife
Let $K$ be a finite, regular cell complex and $f$ be a real valued function on $K$. Then $f$ is called a textit{discrete Morse function} if for all $p$-cell $sigma in K$, the following conditions hold: begin{align*} displaystyle n_{1}=# {tau > sigma mid f(tau)leq f(sigma)} leq 1, \ n_{2}=# {nu < sigma mid f(nu)geq f(sigma)}leq 1. end{align*} A $p$-cell $sigma$ is called a textit{critical $p$-cell} if $n_{1}=n_{2}=0$. A discrete Morse function $f$ is called a textit{perfect discrete Morse function} if the number of critical $p$-cells of $f$ equals to the $p$-th Betti number of $K$ with reference to the coefficient group. The main purpose of this thesis is to compose and decompose perfect discrete Morse functions on connected sums of closed, connected manifolds. We will first discuss the existence of perfect discrete Morse functions on finite complexes and closed, connected, triangulated $n$-manifolds. Secondly, we will show that if the components of a connected sum $M$ of closed, connected, triangulated $n$-manifolds admit a perfect discrete Morse function, then $M$ admits a perfect discrete Morse function that coincides with the perfect discrete Morse functions on the components. Next, we will find a separating sphere on a connected sum $M$ of closed, connected, triangulated surfaces and $3$-manifolds if $M$ admits a perfect discrete Morse function $f$. Finally, we will prove that $f$ can be decomposed as perfect discrete Morse functions on each component of $M$ after some local modifications of it.
Citation Formats
H. Varlı, “Perfect discrete morse functions on connected sums,” Ph.D. - Doctoral Program, 2017.