On Steenrod’s problem for non abelian finite groups. (English) Zbl 0545.57016

Vogel, Pierre

Algebraic topology, Proc. Conf., Aarhus 1982, Lect. Notes Math. 1051, 660-665 (1984).

[For the entire collection see Zbl 0527.00016.]
Let G be a group and X a 1-connected G-space such that \(\tilde H_ iX=0\) for all \(i\neq k\), k some integer \(\geq 2\). We say that (the Moore G-space) X realizes the \({\mathbb{Z}}G\)-module \(\tilde H_ kX\) (in degree k). Steenrod’s problem is whether every \({\mathbb{Z}}G\)-module can so be realized. For certain G (e.g., \(G={\mathbb{Z}}_ p+{\mathbb{Z}}_ p)\), this problem has been resolved in the negative. The author finds an affirmative solution to Steenrod’s problem, however, when \(| G|\) is square free. The arguments use homological techniques and localization.
The reader should be alerted to a small misstatement in the introduction: namely, the author asserts that work of J. Arnold and an obstruction theory of G. Cooke imply an affirmative solution to Steenrod’s problem for all cyclic G. In fact, that argument does not appear to work, although subsequent work of the author (not yet published) does imply the stated conclusion.

Reviewer: P.Kahn

MSC:

57S17	Finite transformation groups
55P99	Homotopy theory
20J06	Cohomology of groups

Keywords:

groups of square free order acting on Moore spaces; spaces with one non trivial homology group; Moore G-space; Steenrod’s problem

Citations:

Zbl 0527.00016