The homology of spaces representing exact pairs of homotopy functors (Q1292692)

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.