Nilpotence and finite H-spaces
A. Zabrodsky defined a finite, connected, associative H-space to be h- nilpotent if one of the following equivalent conditions holds: (1) The functor \([-,X]\) is nilpotent group valued, or (2) for some n, the iterated left commutator map \(X^ n\to X\) is nullhomotopic. The main result of this paper is a characterization of h-nilpotence in terms of (generalized) cohomological criteria. For example, it is shown that X is h-nilpotent if and only if \(K(n)_*X\) is a nilpotent Hopf algebra for all n at all primes p (where K(n) denotes the \(n^{th}\) Morava K-theory at p). Moreover, a corollary of the main theorem states that X is h- nilpotent if its ordinary integral homology is torsionfree. The proof of the main result proceeds by reducing the question of the nullness of the commutator map to the question of the stable nullness of a component of it, and then describing how the nilpotence theorem of \textit{E. S. Devinatz}, \textit{M. J. Hopkins} and \textit{J. H. Smith} [Ann. Math., II. Ser. 128, No.2, 207-241 (1988)] may be applied to yield the required cohomological criteria. The author conjectures that any H-space as above is h-nilpotent, so the cohomological criteria may well be structure results about these spaces.
scientific article; zbMATH DE number 4121019