On the vertex group of n-manifolds

In this paper we obtain the decomposition of the vertex group of n-manifolds, extending the one given by Kauffman and Lins for dimension 3 and solving the related conjecture. The result is obtained in the more general category of gems: the vertex group of a gem Gamma, representing an n-manifold M, is the free product of n copies of the fundamental group of M and a free group F of rank N - n, where N is the number of n-residues of Gamma. In particular, for crystallizations F congruent to Z and consequently the vertex group is an invariant of M.