Invariants and Pictures

This book contains an in-depth overview of the current state of the recently emerged and rapidly growing theory of G_n^k groups, picture-valued invariants, and braids for arbitrary manifolds. Equivalence relations arising in low-dimensional topology and combinatorial group theory inevitably lead to the study of invariants, and good invariants should be strong and apparent. An interesting case of such invariants is picture-valued invariants, whose values are not algebraic objects, but geometrical constructions, like graphs or polyhedra.

In 2015, V O Manturov defined a two-parametric family of groups G_n^k and formulated the following principle: if dynamical systems describing a motion of n particles possess a nice codimension 1 property governed by exactly k particles then these dynamical systems possess topological invariants valued in G_n^k.

The book is devoted to various realisations and generalisations of this principle in the broad sense. The groups G_n^k have many epimorphisms onto free products of cyclic groups; hence, invariants constructed from them are powerful enough and easy to compare. However, this construction does not work when we try to deal with points on a 2-surface, since there may be infinitely many geodesics passing through two points. That leads to the notion of another family of groups — Γ_n^k, which give rise to braids on arbitrary manifolds yielding invariants of arbitrary manifolds.