Manifolds of Triangulations, braid groups of manifolds and the groups $\Gamma_{n}^{k}$

TL;DR

This paper explores groups arising from triangulations of manifolds, introducing new algebraic structures called $\Gamma_{n}^{k}$, which serve as topological and smooth invariants related to Pachner moves and braid groups.

Contribution

It constructs a series of groups $\Gamma_{n}^{k}$ associated with Pachner moves and establishes a canonical map from braid groups to these groups, providing new invariants for manifolds.

Findings

Defined the groups $\Gamma_{n}^{k}$ for triangulation moves

Constructed a canonical map from braid groups to $\Gamma_{n}^{k}$

Provided topological and smooth invariants of manifolds

Abstract

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.