#### Volume 17, issue 1 (2013)

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Combinatorial group theory and the homotopy groups of finite complexes

### Roman Mikhailov and Jie Wu

Geometry & Topology 17 (2013) 235–272
##### 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.

##### Keywords
homotopy groups, braid groups, free product with amalgamation, simplicial groups, spheres, Moore spaces, Brunnian words
##### Mathematical Subject Classification 2010
Primary: 55Q40, 55Q52
Secondary: 18G30, 20E06, 20F36, 55U10, 57M07