Effet d'un attachement cellulaire dans l'homologie de l'espace des lacets. (The effect of attaching cells to the homology of loop spaces)
Main problem: Given a cofibre sequence \(W\to X\to^{f}Y\) (where W \(=\) wedge of spheres), equivalently reformulate the condition that \(\pi_*(\Omega f)\otimes C\) is onto. The classical Milnor-Moore result says that, for coefficients \(C={\mathbb{Q}}\), this happens precisely when \(H_*(\Omega f;C)\) is onto. Further results related to this last condition have recently been obtained by \textit{S. Halperin} and \textit{J.-M. Lemaire} [Math. Scand. 61, No.1, 39-67 (1987; Zbl 0655.55004)]. By considering the associated fibration \(F\to X\to^{f}Y\) and by using their older results on this fibre-cofibre construction [J. Pure Appl. Algebra 53, No.1/2, 59-69 (1988; Zbl 0648.55012)] together with additional Hopf-algebra techniques, the authors generalize here the results of Halperin-Lemaire in two directions: (I) \(C=field\), char \(C\neq Z\) and (II) \(C=subring\) of \({\mathbb{Q}}\), containing 1/2 and 1/3. First main result: The condition \(``H_*(\Omega Y)\) is onto'' is equivalent to (1) \(H_*(\Omega F)\to H_*(\Omega X)\to H_*(\Omega Y)\) is a Hopf exact sequence, (2) \(H_*(\Omega F)\) is a free algebra, (3) the Hopf algebra kernel of \(H_*(\Omega X)\to H_*(\Omega X)/J\), where J is the (Hopf) ideal generated by the Hurewicz images of the attaching maps, is a free algebra whose indecomposables are a free \(H_*(\Omega X/J)\) module, (4) the Poincaré series of \(\Omega X\), \(\Omega Y\) and W are related by \textit{D. J. Anick}'s formula [J. Algebra 78, 120-140 (1980; Zbl 0502.16002)]. Second main result: The condition ``\(H_*(\Omega f)\) is split onto'' is equivalent to (1) \(\pi_*(\Omega f)\) is onto, modulo C-torsion, (2) F is a wedge of spheres, after C-localization. (The coefficients C are used throughout subject to the restrictions (I), respectively (II); in the second result there is also a technical C-decomposability assumption on \(\Omega X\) and \(\Omega Y\), which is automatically satisfied for \(C={\mathbb{Q}}.)\)
scientific article; zbMATH DE number 4124597
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