Note on the homotopy groups of a bouquet $S^1\vee Y$, $Y$ 1-connected

A study is made of the action of the fundamental group of a bouquet of a circle and a 1-connected space on the higher homotopy groups. If the 1-connected space is a suspension space, it is shown, with the aid of a theorem of Hartley on wreath products of groups and the Hilton-Milnor theorem, that the action is residually nilpotent. An unsuccessful approach in the case of a general 1-connected space is discussed, as it has some interesting features.

Keywords

action of fundamental group on higher homotopy groups, residually nilpotent group action, wreath product of groups, Hartley’s theorem, Hilton-Milnor theorem

2010 Mathematics Subject Classification

20E22, 20E26, 55P40, 55Q20

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Published 2 June 2014