Homotopy Torsion Free H-spaces

In [1] Hilton and Reitberg establish a type of homotopy analog of the fact that for any nilpotent group G there is a canonical map 7: G • G/T to a torsion-free nilpotent group which is a rational equivalence (T = torsion subgroup of G). They essentially show that for any nilpotent space X of finite type, there is a map f: X-> X which is a rational equivalence and such that rr,(X) is torsion-free of f'mite type. If one is concerned with H-spaces of finite type the potential for a more complete analogy is even more striking. For, if X is an H-space and W is finite CW, then [W,X] is a (centrally) nilpotent algebraic loop with the set of torsion elements forming a f'mite normal subloop T such that [W,X]/T is torsion-free [3]. With this in mind we will establish the following results: THEOREM A. Let X be an H-space of finite type. Then there exists an H-space X and an H-map f: X->X such that •r,•X) is free abelJan and f is a rational equivalence. COROLLARY B. Let X be an H-space of finite type. Then the X of Theorem A can be taken to be a product of K(Z,n)'s. THEOREM C. Let X be an H-space of finite type and let W be finite CW such that H,(W;Z) is free abelJan. Then there exists an H-space Xw and an H-map f: X w • X which is a rational equivalence and such that (1) [W,Xw] is torsion-free and (2) 0-> [W,Xw] • [W,X] /sexact with finite cokernel. It should be noted that Theorem A is just an extension of ( [ 1], 7.2) to H-spaces and H-maps. Furthermore, since examples abound with H-spaces which have free abelian homotopy groups and which are not products of K(Z,n)'s, Theorem A does not uniquely define a homotopy free analog of an H-space. Corollary B, of course, does given a unique homotopy type deœmed by the type of the H-space. Theorem C is a result that shows that [W,Xw] comes "close " to bein