Geometry & Topology

Combinatorial group theory and the homotopy groups of finite complexes

Roman Mikhailov and Jie Wu

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For n>k3, 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.

Article information

Geom. Topol., Volume 17, Number 1 (2013), 235-272.

Received: 23 September 2011
Revised: 2 October 2012
Accepted: 2 October 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55Q40: Homotopy groups of spheres 55Q52: Homotopy groups of special spaces
Secondary: 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10] 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations 20F36: Braid groups; Artin groups 55U10: Simplicial sets and complexes 57M07: Topological methods in group theory

homotopy groups braid groups free product with amalgamation simplicial groups spheres Moore spaces Brunnian words


Mikhailov, Roman; Wu, Jie. Combinatorial group theory and the homotopy groups of finite complexes. Geom. Topol. 17 (2013), no. 1, 235--272. doi:10.2140/gt.2013.17.235.

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