The Homotopy Groups of a Triad.

In this paper we define new homotopy groups for topological spaces. These groups generalize the homotopy groups of Hurewicz. By the use of these groups and by improved methods we obtain new results about the ordinary homotopy groups, and also easier proofs of known results. Among other things, we can show that 76(S3) is non-trivial. 1. One of the principal problems of modern topology is to devise methods for computing the homotopy groups of a space. The homotopy groups of even such simple spaces as spheres have not been computed except in special cases. This contrasts strongly with the situation for the homology groups, which can be computed for any triangulable space. The homotopy groups resemble homology groups in all basic properties except one: the homology groups are invariant under an excision,' while this is not generally true for the homotopy groups. More precisely, if a space is a union A u B of two subspaces, then under fairly general conditions the inclusion map i: (A, A n B) -* (A u B, B) (called an excision) induces isomorphisms of the corresponding relative homology groups, in all dimensions, but will not generally do so for the relative homotopy groups. This is perhaps the chief reason for the difficulty of computing the homotopy groups. The new homotopy groups defined in this paper are a measure of the deviation from invariance under excision for the relative homotopy groups. We shall use the following notation for certain subsets of cartesian nspace, Cn. The coordinates of a point x e C" are denoted by (xi, ..., n),