Combinatorial group theory and the homotopy groups of finite complexes

Abstract

For $n>k\ge 3$, we construct a finitely generated group with explicit generators and relations obtained from braid groups, whose center is exactly ${\pi }_n\left(S^k\right)$. Our methods can be extended to obtain combinatorial descriptions of homotopy groups of finite complexes. As an example, we also give a combinatorial description of the homotopy groups of Moore spaces.