The homology of spaces representing exact pairs of homotopy functors
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.
representable graded ring valued homotopy functor
scientific article; zbMATH DE number 1307838
scientific article