Consider a pure braid β. It can be expressed as n disjoint strandswhere X(0) = X(1). We assume β to be generic. This means that there are finite number of values. . , m, neither four points of X(t) lie on one circle or line, and the set X(t k ), k = 1, . . . , m contains exactly one quadruple of points which lie on one circle (or line). Then for any regular value t = t k , the set X(t) determines a unique Delaunay triangulation G(t).Fix an integer r ≥ 3. Denote q = e iπ r ∈ C. For any regular value t, consider the set F r (t) = F r (G(t)) of admissible colourings of the edges of the Delaunay triangulation G(t), i.e. maps f : E(G(t)) → N ∪ {0} such that for any triangle of G(t) with edges a, b, c one has

The spaces of triangulations of a given manifolds have been widely studied. The celebrated theorem of Pachner [12] says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves, see also [3,11]. In the present paper we consider groups which naturally appear when considering the set of triangulations with the fixed number of simplices of maximal dimension.There are two ways of introducing this groups: the geometrical one, which depends on the metrics, and the topological one. The second one can be thought of as a "braid group" of the manifold and, by definition, is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version.We construct a series of groups Γ k n corresponding to Pachner moves of (k − 2)-dimensional manifolds and construct a canonical map from the braid group of any k-dimensional manifold to Γ k n thus getting topological/smooth invariants of these manifolds.