Groupoids and Van Kampen's Theorem

Abstract

Introduction The fundamental groupoid TT(X) of a topological space X has been known for a long time but has been regarded, usually, as of little import in comparison with the fundamental group—for example, the groupoid is described in ((3) 155) as an 'interesting curiosity'. In this paper we shall generalize the fundamental group at a point a of X, namely 7r(X,a), to the fundamental groupoid on a set A, written TT(X, A), which consists of the homotopy classes of paths in X joining points of A n X. If a e A, the fundamental group TT{X,O) can be recovered from knowledge of n(X,A), but the latter groupoid is often easier to describe because the category of groupoids has exactly the right properties to model successfully the geometric constructions in building up spaces. As an example of this, consider an adjunction space Bu^Z, where / : Y -> B is continuous and Y is a closed subspace of Z (so that B\JfZ is obtained by glueing Z to B by means of/) . The following is a basic step in computing the fundamental group of a space: Compute the fundamental group of B\JfZ in terms of the fundamental groups of B, Y, Z and the maps induced by i: Y -» Z, f: Y -y B. This is insoluble without some local conditions on Y in Z, as an example of H. B. Griffiths (6) shows. With suitable local conditions (and other topological conditions which may be regarded as inessential) a special case of this problem was solved by van Kampen in (10). His answer was a formula describing the fundamental group of B \JfZ in terms of generators and relations. We shall use groupoids to give in Theorem 4.2 a general and natural solution to this problem. With this theorem one can derive by a uniform method the fundamental groups of spaces from a large class which includes all CW-complexes, and, a fortiori, all simplicial complexes. Even for the latter spaces the methods here are simpler than the classical combinatorial methods. Theorem 4.2 is a deduction from our main result, Theorem 3.4, which determines a groupoid TT(X, AO) when X is the union of the interiors of two sets Xlt X2. Other work on the fundamental group of a union has been done by P. Olum (11), the author (1), R. H. Crowell (2), and A. I. Weinzweig (12). The results of (11) and (1) are not as powerful