  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.
