Combinatorial group theory and the homotopy groups of finite complexes

Mikhailov, Roman¹ ; Wu, Jie²

¹ St Petersburg Department of Steklov Mathematical Institute, and, Chebyshev Laboratory, St Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178 Russia
² Department of Mathematics, National University of Singapore, 2Block S17-06-02, 10 Lower Kent Ridge Road, Singapore 119076, Singapore

Geometry & topology, Tome 17 (2013) no. 1, pp. 235-272.

Résumé

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.

Détail

DOI : 10.2140/gt.2013.17.235

Classification : 55Q40, 55Q52, 18G30, 20E06, 20F36, 55U10, 57M07
Keywords: homotopy groups, braid groups, free product with amalgamation, simplicial groups, spheres, Moore spaces, Brunnian words

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