Universal Abelian H-spaces (Q649816)

A universal Abelian H-space for a space \(X\) is an Abelian H-space \(T(X)\) together with a map \(X \rightarrow T(X)\) such that any map from \(X\) to an Abelian H-space \(Z\) extends, uniquely up to homotopy, to an H-map \(T(X) \rightarrow Z\). As one might expect this kind of object is rather rare (this reviewer does not think that it exists for Eilenberg-Mac Lane spaces), but known examples include \(T(S^{2n}) = \Omega S^{2n+1}\), localized at an odd prime. Also, the ``Abelianization'' of a Moore space \(P^{2n+1}(p^r)\) is the Anick space \(T_{2n}(p^r) = S^{2n+1} \{ p^r \}\), homotopy fiber of the degree \(p^r\) map on the sphere \(S^{2n+1}\). However the features of the odd Anick spaces \(T_{2n-1}(p^r)\) are not fully understood and the results in this article contrast with \textit{S. D. Theriault}'s [Trans. Am. Math. Soc. 353, No. 3, 1009--1037 (2001; Zbl 0992.55008)]. The author proves for example that the torsion exponent of the Abelianization must be equal to that of the original space, and that odd dimensional homology groups must be torsion free if one starts with a space with bounded torsion and trivial \(p\)-th powers in mod \(p\) cohomology. Therefore an odd Anick space cannot be the universal Abelian H-space for a Moore space \(P^{2n}(p^r)\). In fact, it is not known whether the Anick spaces are homotopy commutative H-spaces. As for the universal property, some weaker features are investigated and one could replace maps into all Abelian H-spaces with maps into double loop spaces, or loop spaces on so-called subexponential spaces.