  The spaces of triangulations of a given manifold 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 fixed number of simplices of maximal dimension. There are
three ways of introducing this groups: the geometrical one, which depends on
the metric, the topological one, and the combinatorial 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.
