# Transformation of the fundamental group of Lie groups under group homomorphisms

Hello!

I have encountered the following problem while trying to solve a different one: let G, H be two semisimple Lie groups, and $G\to H$ a Lie group homomorphism. Does anyone know if the question of how the fundamental group of G maps to the fundamental group of H has been studied in some detail? In fact, I need to know how this works when H is a real form of a Lie group and G is a maximal split form contained in it.

Thanks!

One way to approach both the general and the specialized questions you raise is to study these Lie groups as $\mathbb{R}$-points of complex semisimple algebraic groups defined over $\mathbb{R}$. In this framework there is a lot of relevant work on both structure and homomorphisms by Borel and Tits, following their foundational 1965 paper on reductive groups. Fundamental groups are here characterized algebraically in terms of the position of the character group of a maximal torus inside the full abstract group of weights for the given root system.