Quillen spectral sequences in rational homotopy theory (Q2708734)

In the fundamental paper on rational homotopy theory [Rational homotopy theory, Ann. Math., II. Ser. 90, 205-295 (1969; Zbl 0191.53702)], \textit{D. Quillen} introduced a spectral sequence that relates the rational homotopy Lie algebra, \({\mathcal L}_*(X) = \pi_*(\Omega X)\otimes \mathbf{Q}\) with the Samelson product of the constituent spaces in a cofibration \(A \to X \to C\). This spectral sequence has \(E^2\)-term given by \(E^2_{p,q} \cong {\mathcal L}_p(A)\amalg {\mathcal L}_q(C)\) where \(\amalg\) denotes the coproduct in the category of connected graded \textbf{Q}-Lie algebras and the spectral sequence of Lie algebras converges to \({\mathcal L}_*(X)\). This result may be understood as the Eckmann-Hilton dual of the Serre spectral sequence of homology coalgebras of spaces in a fibration. Using the Adams cobar construction, the author adds a spectral sequence of the type introduced by Quillen for a cofibration \(A\to X\to C\) for which the \(E^2\)-term is given by \(E^2_{p,q} \cong H_p(\Omega A; k)\amalg H_q(\Omega C; k)\) where \(\amalg\) denotes the coproduct in the category of algebras over a field \(k\) and the first quadrant spectral sequence of algebras converges to \(H_*(\Omega X;k)\). The construction is based on a filtration of the cobar complex. In the case of cocommutative normalized chain complexes, the spectral sequence is a spectral sequence of Hopf algebras where the coproduct is in the category of homology Hopf algebras. In contrast to the Serre spectral sequence, the dual version of the Hirsch conjecture is true for the spectral sequence of homology algebras, that is, if \(A\to X \to C\) is a cofibration of nicely pointed spaces, and the mapping \(X\to C\) induces an epimorphism \(H_*(\Omega X;k) \to H_*(\Omega C;k)\), then the spectral sequence collapses at \(E^2\) and, as algebras, NEWLINE\[NEWLINEH_*(\Omega X;k) \cong H_*(\Omega A;k)\amalg H_*(\Omega C;k).NEWLINE\]NEWLINE Finally, the author recovers Quillen's spectral sequence of Lie algebras using the homology spectral sequence and the theorem of Cartan-Serre and Moore that, when \(X\) is an H-space, the Lie algebra of primitives in \(H_*(X;\mathbf{Q})\) gives \(\pi_*(X)\otimes \mathbf{Q}\).NEWLINENEWLINEFor the entire collection see [Zbl 0949.00018].