Manifolds of Triangulations, Braid Groups of Manifolds, and the Groups Γnk

Abstract

The spaces of triangulations of a given manifold have been widely studied. The celebrated theorem of Pachner (Eur J Comb 12:129–145, 1991) says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves, or Pachner moves, see also Gelfand et al. (Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser, Boston (1994)), Nabutovsky (Comm Pure Appl Math 49:1257–1270, 1996). 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 groupsΓnk corresponding to Pachner moves of (k − 2)-dimensional manifolds and construct a canonical map from the braid group of any k-dimensional manifold toΓnk thus getting topological/smooth invariants of these manifolds.