#### 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
##### Publication
Received: 23 September 2011
Revised: 2 October 2012
Accepted: 2 October 2012
Published: 7 March 2013
Proposed: Haynes Miller
Seconded: David Gabai, Bill Dwyer
##### Authors
 Roman Mikhailov St Petersburg Department of Steklov Mathematical Institute and Chebyshev Laboratory St Petersburg State University 14th Line, 29b Saint Petersburg 199178 Russia http://www.mi.ras.ru/~romanvm/ Jie Wu Department of Mathematics National University of Singapore 2Block S17-06-02, 10 Lower Kent Ridge Road Singapore 119076 Singapore http://www.math.nus.edu.sg/~matwujie