On the Connection between the Second Relative Homotopy Groups of Some Related Spaces

The title of this paper is chosen to imitate that of the paper by van Kampen [10] which gave some basic computational rules for the fundamental group TTX{ Y, £) of a based space (an earlier more special result is due to Seifert [14]). In [1] results more general than van Kampen's were obtained in terms of fundamental groupoids. The advantage of the use of groupoids is that one obtains an easy description of the fundamental groupoid of a union of spaces even when the spaces and their intersections are not pathconnected ; in such cases, the computation of the fundamental group is greatly simplified by using groupoids. To obtain analogous results in dimension 2 we make essential use of a kind of double groupoid first described in [4]. A major aim is to introduce the homotopy double groupoid p(X, Y,Z) defined for any triple (X, Y,Z) of spaces such that every loop in Z is contractible in Y. The methods of [1] are generalized to give results on p(X, Y,Z). We obtain, as algebraic consequences, results on the second relative homotopy group 7T2{X, Y, t) in the form of computational rules for the crossed module