Fundamental group of homotopy colimits

We give a general version of theorems due to Seifert-van Kampen and Brown about the fundamental group of topological spaces. We consider here the fundamental group of a general homotopy colimit of spaces. This includes unions, direct limits and quotient spaces as special cases. The fundamental group of the homotopy colimit is determined by the induced diagram of fundamental groupoids via a simple commutation formula. We use this framework to discuss homotopy (co-)limits of groups and groupoids as well as the useful Classification Lemma 6.4. Immediate consequences include the fundamental group of a quotient spaces by a group action ̃π₁(K/G) and of more general colimits. The Bass-Serre and Haefliger's decompositions of groups acting on simplicial complexes is shown to follow effortlessly. An algebraic notion of the homotopy colimit of a diagram of groups is treated in some detail.