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We consider the Belkale-Kumar cup product ⊙ t \odot _t on H ∗ ( G / P ) H^*(G/P) for a generalized flag variety G / P G/P with parameter t ∈ C m t \in \mathbb {C}^m , where m = dim ⁡ ( H 2 ( G / P ) ) m=\dim (H^2(G/P)) . For each t ∈ C m t\in \mathbb {C}^m , we define an associated parabolic subgroup P K ⊃ P P_K \supset P . We show that the ring ( H ∗ ( G / P ) , ⊙ t ) (H^*(G/P), \odot _t) contains a graded subalgebra A A isomorphic to H ∗ ( P K / P ) H^*(P_K/P) with the usual cup product, where P K P_K is a parabolic subgroup associated to the parameter t t . Further, we prove that ( H ∗ ( G / P K ) , ⊙ 0 ) (H^*(G/P_K), \odot _0) is the quotient of the ring ( H ∗ ( G / P ) , ⊙ t ) (H^*(G/P), \odot _t) with respect to the ideal generated by elements of positive degree of A A . We prove the above results by using basic facts about the Hochschild-Serre spectral sequence for relative Lie algebra cohomology, and most of the paper consists of proving these facts using the original approach of Hochschild and Serre.

In a fibration OmegaF (Omegaj)under right arrow OmegaX (Omega pi )under right arrow OmegaB we show that finiteness conditions on F force the homology Serre spectral sequence with F-p-coefficients to collapse at some finite term. This in particular implies that as graded vector spaces, H-*(OmegaX) is "almost" isomorphic to H-*(OmegaB) circle times H-*(OmegaF). One consequence is the conclusion that X is elliptic if and only if B and F are.

Using methods developed by W. Singer and J. P. May, we describe a systematic approach to showing that many spectral sequences, determined by a filtration on a complex whose homology has an action of operations, possess a compatible action of the same operations. As a consequence, we obtain W. Singer’s result for Steenrod operations on Serre spectral sequence and extend A. Bahri’s action of Dyer-Lashof operations on the second quadrant Eilenberg-Moore spectral sequence.

The purpose of this paper is to show that transfers for fibrations behave well with respect to spectral sequences which are induced by filtrations of the base space. In particular, for the spectral sequence of a fibration (induced by the skeletal filtration of the base space), one obtains the expected effect on the E 2 {E_2} -terms: We prove that the transfer in the E 2 {E_2} -terms is determined by the transfer of the fiber (considered trivially as a fibration over a point). As an application, results of Atiyah on the K K -theory of classifying spaces are transcribed to generalized cohomology theories.

It is proved that if G is a finite group, H a normal subgroup, and A a finitely generated G-module, then both the cohomology and homology spectral sequences for the group extension stop in a finite number of stops. A lemma about Tor(M, N) as a module over R 0 S is proved. Two spectral sequences of Hochschild and Serre are shown to be