Sectionable Tournaments: Their topology and Coloring

We provide a detailed study of topological and combinatorial properties of sectionable tournaments. This class forms an inductively constructed family of tournaments grounded over simply disconnected tournaments, those tournaments whose fundamental groups of acyclic complexes are non-trivial. When $T$ is a sectionable tournament, we fully describe the cell-structure of its acyclic complex $Acy(T)$ by using the adapted machinery of discrete Morse theory for acyclic complexes of tournaments. In the combinatorial side, we demonstrate that the dimension of the complex $Acy(T)$ has a role to play. We prove that if $T$ is a $(2r+1)$-sectionable tournament and $d$ is the dimension of $Acy(T)$, then the (acyclic) chromatic number of $T$ satisfies $\chi(T)\leq 2 \left( 2-1/(r+1) \right)^{\log(d+1)}-1$ where the logarithm has two as its base.