The Serre spectral sequence of a multiplicative fibration (Q2716162)

In this well written paper the authors analyze systematically the homology Serre spectral sequence of the loop space fibration \(\Omega F\to \Omega X\to \Omega B\) obtained by looping a fibration of simply connected pointed spaces \(F\), \(X\) and \(B\) of finite type. They show that in the presence of certain finiteness conditions on the fiber \(F\) the homology Serre spectral sequence over a prime field of positive characteristic \(p\) collapses at some finite stage and so the space \(\Omega X\) is closely related to the product space \(\Omega F\times \Omega B\). The homology Serre spectral sequence of the loop space fibration was first studied in detail by \textit{W. Browder} [ibid. 107, 153-176 (1963; Zbl 0114.39304)], who showed for example, that one gets in fact a spectral sequence of cocommutative Hopf algebras. Another systematic, but different approach to investigate multiplicative fibrations and the mod \(p\) homology groups of spaces in loop space fibrations using the Eilenberg-Moore spectral sequence was undertaken by \textit{J. C. Moore} and \textit{L. Smith} in their fundamental papers [Am. J. Math. 90, 752-780 (1968; Zbl 0194.24501); 1113-1150 (1968; Zbl 0214.50201)].NEWLINENEWLINENEWLINEIn the paper under review the authors follow a different approach, which allows them to analyze fairly general fibrations without loosing the Hopf algebra structure in the terms of the Serre spectral sequence. Namely, under the assumption that the Hopf centre of the Hopf algebra \(H_*(\Omega X;\mathbb{F}_p)\), the union of all its central sub Hopf algebras, is a finitely generated algebra, they show that the Serre spectral sequence collapses and gives an isomorphism of graded vector spaces NEWLINE\[NEWLINEH_*(\Omega X; \mathbb{F}_p)\otimes A\cong H_*(\Omega F;\mathbb{F}_p)\otimes H_*(\Omega B;\mathbb{F}_p) \otimes \Lambda VNEWLINE\]NEWLINE where \(A\) is a tensor product of finitely many monogenic Hopf algebras and \(\Lambda V\) is an exterior algebra on a finite dimensional vector space concentrated in odd degrees. This follows by analyzing carefully the Hopf algebra structure through all the stages of the spectral sequence, which constitutes the main technical part (\S 6 and \S 7) of the paper. Instead of using Browder's biprimitive spectral sequence, they show that each term \(E^r\) of the Serre spectral sequence can be written as Hopf algebra in the form \(E^r_{*, 0}\otimes\Lambda U^r\otimes E^r_{0,*}\), where \(\Lambda U^r\) is the exterior Hopf algebra on the vector space \(U^r\) of primitive elements. This gives also a refinement of Browder's classical result and allows to study the differentials \(d^r\) directly through the Hopf algebra structure of the spectral sequence.NEWLINENEWLINENEWLINEA lot of examples and applications (\S 5) are given. The authors show for example, that the Serre spectral sequence of a loop fibration of a principal fibration with fiber \(K(\Gamma,n)\) collapses at some finite stage, if \(H_*(B)\) is finite dimensional. Another application is given for elliptic spaces, i.e. spaces such that for each prime \(q\) their loop space homology \(H_*(\Omega Y; \mathbb{F}_q)\) grows polynomially. This important class of spaces was introduced by the authors in [Bull. Am. Math. Soc., New Ser. 25, No. 1, 69-73 (1991; Zbl 0726.55006)]. In the paper under review they verify now that, if \(F,X\) and \(B\) are simply connected spaces of finite type, then it follows that \(X\) is elliptic if and only if \(F\) and \(B\) are elliptic.