La successió espectral de Serre i algunes aplicacions

Abstract

[en] The aim of this work is to introduce Serre’s spectral sequence. Spectral sequences are a very powerful tool that allows us to relate the homology (or cohomology) groups of various topological spaces when we cannot do so using other simpler methods such as exact couples. The basic idea is to calculate successive approximations of the invariant we want to find, so that each term increases the level of precision, until we obtain it in the most favorable cases. However, its great utility implies an increase in the difficulty of the tools used, mostly based on homological algebra. In our case, Serre’s spectral sequence allows us to relate the homology (or cohomology) groups of the base, fiber, and total space of a fibration, under some hypotheses about the structure of the base. Finally, the possibility of building a fibration from any space, called path fibration, will open up a wide range of possibilities for applying Serre’s spectral sequence.