O grupo fundamental de variedades flag reais

Abstract

The objective of this work is to present two approaches for computing the fundamental group of a flag manifold FΘ = G/PΘ. The first approach is based on [1] and involves calculating the fundamental group of a flag manifold via generators and relations, using the cellular structure obtained from the Bruhat decomposition and the van Kampen Theorem (see Theorem 2.25). The second approach, based on [2], relies on a result from covering theory, which states that the fundamental group of G/PΘ is isomorphic to a quotient of a special discrete subgroup (see Theorem 3.1). More specifically, it is a quotient of the group of connected components of PΘ (see Proposition A.65). In this work, we use this result to explore in detail the computation of the fundamental group of the flag manifolds of SL(n, R). This second approach has a more algebraic nature, as the universal covering of SO(n, R) is described through Clifford algebras.