Nilpotence and finite H-spaces
type: Item
instance of: scholarly article
title: Nilpotence and finite H-spaces
zbMATH Open document ID: 0684.55007
DOI: 10.1007/BF02765895
author: Michael J. Hopkins
published in: Israel Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 55P45
Mathematics Subject Classification ID: 55P42
zbMATH DE Number: 4121019
zbMATH Keywords: stable homotopy
zbMATH Keywords: associative H-space
zbMATH Keywords: h-nilpotent
zbMATH Keywords: nilpotence theorem
reviewed by: John Oprea
MaRDI profile type: Publication
cites work: Nilpotence and stable homotopy theory. I
cites work: Nilpotence and stable homotopy theory. II
cites work: Q3747338
cites work: Q4107640
full work available at URL: https://doi.org/10.1007/bf02765895
OpenAlex ID: W2064389184
description: scientific article; zbMATH DE number 4121019