Combinatorial group theory and the homotopy groups of finite complexes
The calculation of homotopy groups of spheres and Moore spaces is a key problem in algebraic topology. \textit{J. Wu} in the paper [Math. Proc. Camb. Philos. Soc. 130, No. 3, 489--513 (2001; Zbl 0986.55013)] obtains a combinatorial description of the homotopy groups of \(S^2\). \textit{F. R. Cohen} and \textit{J. Wu} [Q. J. Math. 62, No. 4, 891--921 (2011; Zbl 1252.55005)] continue this analysis in a way which leads to the present paper. Using Milnor's FK construction, braid groups, free products with amalgamation, simplicial theory, etc., the authors obtain remarkable descriptions of the homotopy groups of spheres as the centers of explicitly constructed groups. They also obtain such descriptions of \(\pi_n\) of Moore spaces when \(n\neq 3\). As one might expect, the paper uses a considerable amount of combinatorial group theory. As a minor quibble, the reviewer would expect a reference (out of a total of 26) to [\textit{E. H. Brown jun.}, Ann. Math. (2) 65, 1-20 (1957; Zbl 0077.16804)].
scientific article; zbMATH DE number 6152262
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