Geometry & Topology
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 Geom. Topol., 2013, Volume 17, Issue 1, Pages 235–272 (Mi gt2)

Combinatorial group theory and the homotopy groups of finite complexes

R. Mikhailovab, J. Wuc

a Chebyshev Laboratory, St Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178 Russia
b St. Petersburg Department of Steklov Mathematical Institute
c Department of Mathematics, National University of Singapore, 2Block S17-06-02, 10 Lower Kent Ridge Road, Singapore 119076, Singapore

Abstract: For $n>k\geqslant3$, we construct a finitely generated group with explicit generators and relations obtained from braid groups, whose center is exactly $\pi_n(S^k)$. 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.

 Funding Agency Grant Number National Natural Science Foundation of China 11028104 Ministry of Education and Science of the Russian Federation 11.G34.31.0026 Ministry of Education, Singapore R-146-000-137-112R-146-000-143-112 This article was finished during the visit of both authors to Dalian University of Technology under the support of a grant (No. 11028104) of NSFC of China in July of 2011. The authors would like to thank the hospitality of Dalian University of Technology for supporting our research on this topic. The authors are thankful to L Breen, H Miller, S Theriault and the anonymous referee for their valuable comments and suggestions to improve the manuscript. The research of the first author is supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government grant 11.G34.31.0026 and the research of the second author is supported in part by the AcRF Tier 1 (WBS No. R-146-000-137-112) and AcRF Tier 2 (WBS No. R-146-000-143-112) of MOE of Singapore and a grant (No. 11028104) of NSFC of China.

DOI: https://doi.org/10.2140/gt.2013.17.235

Bibliographic databases:

MSC: Primary 55Q40, 55Q52; Secondary 18G30, 20E06, 20F36, 55U10, 57M07
Revised: 02.10.2012
Accepted:02.10.2012
Language: