From Kalai's classic paper generalizing Cayley's tree-enumeration formula to simplicial complexes  it is known that simplicial complexes on a small number of vertices can have enormous torsion in homology Moreover  in a random setting one may find instances of this phenomenon such as  for example  a 3-dimensional simplicial complex on 30 vertices with the torsion subgroup of the second homology group having order larger than 10^82