Manifolds of Triangulations, Braid Groups of Manifolds, and the Groups $$\Gamma _{n}^{k}$$

The spaces of triangulations of a given manifolds have been widely studied. The celebrated theorem of Pachner \cite{Pachner} says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves, see also \cite{GKZ,Nabutovsky}. 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 $\Gamma_{n}^{k}$ corresponding to Pachner moves of $(k-2)$-dimensional manifolds and construct a canonical map from the braid group of any $k$-dimensional manifold to $\Gamma_{n}^{k}$ thus getting topological/smooth invariants of these manifolds.