Càlcul de la cohomologia de grups de Lie amb successions espectrals

Abstract

The aim of this work is to use the Serre spectral sequence to calculate the cohomology of the classical Lie groups and their classifying spaces. In the development of algebraic topology during the decades of 1920 and 1930, new homotopy invariants were introduced, namely homotopy groups and cohomology groups. Their predecessors were the homology groups and the fundamental group, which is the first homotopy group. Computing these invariants can be a complicated issue, especially in the case of homotopy groups, for which there is no efficient calculation method. This is why some tools for relating all these invariants began to appear soon. Important examples were the Hurewicz theorem, which relates the homotopy groups and the homology groups of a space; the Künneth formula for the homology of the product of two spaces, or the long exact sequence associated to a fibration, which relates the homotopy groups of the three spaces involved in the fibration. The latter one posed a natural question: how are the homologies or cohomologies of the three spaces in a fibration related? A reason to answer this question was that the inclusion of a closed subgroup $H$ in a topological group defines a fibration $H \rightarrow G \rightarrow G/H$. The answer to this question is that the homology and cohomology groups of the speces in a fibration are related by a spectral sequence. The first step toward this answer was taken by J. Leray [7] in 1946, but the development of this machinery occurred in 1950 with Serre [11]. Since then, other kinds of spectral sequences, relating other homotopy invariants, have been introduced. However we will focus only on the one mentioned above, which is called Serre spectral sequence. Among the great successes of the Serre spectral sequence there is a proof of the finiteness of the homotopy groups of spheres, the calculation of the homology and cohomogy groups of loop spaces, and of the homologies and cohomologies of the Lie groups and their classifying spaces. As we mentioned above, the structure of Lie groups leads in a natural way to use this tool to calculate their homology and cohomology groups. An example is given by the fibration $U(n−1) \rigtarrow U (n) \rightarrow S^{2n−1}$ , which is used in Section 2.6 to calculate the cohomology of $U(n)$. Furthermore, the group product induces an extra structure into the cohomology ring and the homology modules which restricts the possible cohomologies and homologies of these groups. The study of the cohomology of Lie groups began by the end of the decade of 1920 with E. Cartan, and it continued by, among many others, Pontrjagin and Hopf. All these authors studied cohomology with real coefficients. The introduction of the Serre spectral sequence provided a new way to calculate the homology and cohomology of Lie groups over the integers or $\mathbb {Z}/p$. The principal contribution to these calculations was due to A. Borel [1], who also gave important results on the cohomology of the classifying spaces of Lie groups. Classifying spaces of Lie groups classify principal bundles and vector bundles over a space $X$ by means of the homotopy classes of maps from $X$ to the classifying space. This makes the cohomology of classifying spaces a relevant issue in the classification of fibre bundles. In the first chapter we explain the concepts of fibre bundle, vector bundle and fibration, and we mention a few basic results. We remark the relevance of a particular fibration: the path-space fibration. Then we introduce Eilenberg–MacLane spaces and Postnikov towers. In the following sections we explain the construction and usage of the Serre spectral sequence. Finally, in the last section we show some applications. In particular we calculate the cohomologies of some Eilenberg–Mac Lane spaces and use them to prove the finiteness of the homotopy groups of the spheres. In the second chapter we begin by introducing briefly topological groups and Lie groups, and in particular the classical compact Lie groups. Next we define principal bundles and the classifying space of a topological group, and we describe the classifying spaces of the unitary and orthogonal groups. Then we define Hopf algebras and characterize the Hopf algebras over the rationals. In the next section we see that the group product induces the structure of Hopf algebra into the cohomology ring, and we use the characterization above to deduce the rational cohomology of classical groups. Next we relate the cohomology over a field of a topological group with the cohomology of its classifying space. In the next sections we calculate the integer cohomologies of the unitary group and its classifying space, and the $\mathbb{Z}/2$ cohomology of the classifying space of the orthogonal group. Finally, we use these results to introduce the Chern classes associated to a complex vector bundle and the Stiefel–Whitney classes associated to a real vector bundle.