The homology of spaces representing exact pairs of homotopy functors
type: Item
instance of: scholarly article
title: The homology of spaces representing exact pairs of homotopy functors
zbMATH Open document ID: 0929.55007
DOI: 10.1016/S0040-9383(98)00037-8
author: John Hunton
author: Paul Turner
published in: Topology
publication date: 2000-01-25T00:00:00Z
review text: The main result of the article (Theorem 3.5) asserts the following. Let \({\mathcal R}\) be a representable graded ring valued homotopy functor and let \({\mathcal M}\) be a representable graded \({\mathcal R}\) module functor. Assume that the ring and the module functors \({\mathcal R}\) and \({\mathcal M}\) are exact; that is, \({\mathcal M}(X)={\mathcal R}(X)\otimes_{{\mathcal R}(pt)}{\mathcal M}(pt)\) for any finite space \(X\). Then there exists an isomorphism of coalgebraic modules \(H_* (\underline M_*) \cong H_* (\underline R_*)\overline \otimes_{\mathbb{Z}/ p[R^*]} \mathbb{Z}/p[M^*]\). In the case where \({\mathcal R}\) is complex cobordism and \({\mathcal M}\) is a Landweber exact cohomology theory, the result goes back to \textit{M. J. Hopkins} and \textit{J. R. Hunton} [ibid. 34, No. 1, 29-36 (1995; Zbl 0862.55005)]. The authors point out that their main result has a number of generalizations. The main technique used by the authors is to examine the homology of the representing spaces by a colimit construction over suitable detecting categories. This is done in the line of \textit{D. Quillen}'s work on the cohomology of groups [Ann. Math., II. Ser. 94, 549-572, 573-602 (1971; Zbl 0247.57013)]. The authors give some applications, in particular, to the homology of \(\Omega\) spectra, and point out potentially important aspects of the technique which go beyond the results of their article.
Mathematics Subject Classification ID: 55N99
Mathematics Subject Classification ID: 55N10
Mathematics Subject Classification ID: 18G99
Mathematics Subject Classification ID: 55R40
zbMATH DE Number: 1307838
zbMATH Keywords: representable graded ring valued homotopy functor
zbMATH Keywords: \({\mathcal R}\)-module functor
zbMATH Keywords: exact pair of functors
reviewed by: Krzysztof Pawałowski
MaRDI profile type: Publication
full work available at URL: s0040-9383(98)00037-8
OpenAlex ID: W2061838659
Wikidata QID: Q128081702
Recommended article: Generalized André-Quillen cohomology
Recommended article: Operations in equivariant ℤ/p-cohomology
Recommended article: Cohomology of topologised monoids
Recommended article: Q3997902
Recommended article: Q3125635
Recommended article: Brown representability for the dual
Recommended article: Q5692782
Recommended article: Spectral duality for block cohomology
Recommended article: Equivariant coarse (co-)homology theories
Recommended article: The tower of n-groupoids and the long cohomology sequence
description: scientific article; zbMATH DE number 1307838