Combinatorial group theory and the homotopy groups of finite complexes

A description of homotopy groups of the 2-dimensional sphere in terms of combinatorial group theory was discovered by the second author in 1994 and given in his thesis [25], with a published version in [27]. In this article we give a combinatorial description of the homotopy groups of k-dimensional spheres with k ≥ 3. The description is given by identifying the homotopy groups as the center of a quotient group of the self free products with amalgamation of pure braid groups by certain canonical subgroups (Theorem 2.2, see Section 2 for the explicit construction). Our methods can be extended for obtaining combinatorial descriptions of homotopy groups of finite complexes. As an example, we also give a combinatorial description of the homotopy groups of Moore spaces.