Volume 17, issue 1 (2013)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 3, we construct a finitely generated group with explicit generators and relations obtained from braid groups, whose center is exactly πn(Sk). 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
References
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