Geom. Topol., 2013, Volume 17, Issue 1, Pages 235–272
Combinatorial group theory and the homotopy groups of finite complexes
R. Mikhailovab, J. Wuc
a Chebyshev Laboratory,
St Petersburg State University,
14th Line, 29b,
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,
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.
|National Natural Science Foundation of China
|Ministry of Education and Science of the Russian Federation
|Ministry of Education, Singapore
|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.
MSC: Primary 55Q40, 55Q52; Secondary 18G30, 20E06, 20F36, 55U10, 57M07
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