Leray-Serre Spectral Sequence and Some of its Applications

Abstract

The purpose of this thesis is to look at how the Leray-Serre spectral sequence is constructed and explore some of their applications. For the purpose of this thesis, our spectral sequences will be based on filtered complexes, however one can use exact couples. Spectral sequences have been one of the greatest achievements of Algebraic Topology since their invention by Leray in the late 1940’s, hence the name Leray-Serre spectral sequences. Ever since, they have become a very useful tool in many disciplines like Geometry, Topology and Algebra. For example, for a fibration, spectral sequences play an important role in identifying how the cohomology rings of the total space E, the base space B and the fibre F are related. We discuss the construction in brief and give its applications in computing the cohomology algebra of unitary groups, Stiefel and infinite dimensional complex Grassmannian manifolds, over Q.