Abstract
We prove that the fundamental group and the integral homology groups of a poset with fewer than 13 points are torsion free, settling a conjecture of Hardie, Vermeulen and Witbooi and answering a question of Barmak. In addition, we prove that if a poset has fewer than 16 points then the geometric realization of its order complex can not be homotopy equivalent to either the torus or the Klein bottle, answering another open question. Furthermore, we find all the posets of 16 points (resp. of 13 points) such that the geometric realizations of their order complexes are homotopy equivalent to either the torus or the Klein bottle (resp. to the real projective plane).
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