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A construction of p-local H-spaces. (English) Zbl 0582.55010
Algebraic topology, Proc. Conf., Aarhus 1982, Lect. Notes Math. 1051, 351-359 (1984).
[For the entire collection see Zbl 0527.00016.]
Let X be the localization at a prime p of a complex \(S^{n_ 1}\cup e^{n_ 2}\cup...\cup e^{n_{\ell}}\) with \(n_ i\leq n_{i+1}\) and each \(n_ i\) odd. The main result in this paper is the following: If \(\ell <p-1\), then there is a p-local H-space M(X) and a map \(X\to M(X)\) such that \(H_*M(X)\) is the exterior algebra generated by the injection of \(\tilde H_*(X)\). This relation between the complex X and the H-space M(X) reminds one of the connection between the quasiprojective spaces of James and the classical Lie groups. As an existence theorem, this result gives an independent construction of the low rank torsion-free H-spaces first given by G. Cooke, J. R. Harper and A. Zabrodsky [Topology 18, 349-359 (1979; Zbl 0426.55009)].
The authors do not build M(X) directly. Instead they obtain it as a retract of \(\Omega\) \(\Sigma\) X by constructing an appropriate complement for it. In more detail, let L denote the free Lie algebra generated by \(\tilde H_*X\). Then \(H_*\Omega \Sigma X\) is the universal enveloping algebra U L. The authors produce a space \(\lambda\) (X) and a space \(\theta\) : \(\lambda\) (X)\(\to \Sigma X\) such that \(\Omega \theta_*\) maps \(H_*(\Omega \lambda (X))\) isomorphically onto U [L,L] in U L. This is the key step. It requires detailed homological information about the loop spaces involved as well as some representation theory of the symmetric group.
Now define M(X) to be the homotopy fiber of \(\theta\) and consider the corresponding fibration \(\Omega\) \(\lambda\) (X)\(\to \Omega \Sigma (X)\to M(X)\). From homological properties of this fibration they deduce that \(\Sigma\) M(X) contains \(\Sigma\) X as a retract. This in turn provides them with a section.
The only flaw in this interesting paper is that pages 355 and 356 appear in the wrong order.