  Combinatorially and stochastically defined simplicial complexes often have
the homotopy type of a wedge of spheres. A prominent conjecture of Kahle
quantifies this precisely for the case of random flag complexes. We explore
whether such properties might extend to graphs arising from nature. We consider
the brain network (as reconstructed by Varshney & al.) of the Caenorhabditis
elegans nematode, an important model organism in biology. Using an iterative
computational procedure based on elementary methods of algebraic topology,
namely homology, simplicial collapses and coning operations, we show that its
directed flag complex is homotopy equivalent to a wedge of spheres, completely
determining, for the first time, the homotopy type of a flag complex
corresponding to a brain network.
  We also consider the corresponding flag tournaplex and show that torsion can
be found in the homology of its local directionality filtration. As a toy
example, directed flag complexes of tournaments from McKay's collection are
classified up to homotopy. Moore spaces other than spheres occur in this
classification. As a tool, we prove that the fundamental group of the directed
flag complex of any tournament is free by considering its cell structure.

Algebraic Topology