Artin's braids, braids for three space, and groups σ _n ⁴ and G _n ^k

We construct a group σn4 corresponding to the motion of points in R ³ from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on n strands to the product of copies of σn4. We will also study the group of pure braids in R ³, which is described by a fundamental group of the restricted configuration space of R ³, and define the group homomorphism from the group of pure braids in R ³ to σn4. At the end of this paper, we give some comments about relations between the restricted configuration space of R ³ and triangulations of the 3-dimensional ball and Pachner moves.