Homotopy Groups of Certain Algebraic Systems

We shall describe an algebraic scheme giving rise to a group P2 that can be identified as the second homotopy group when the elements entering the scheme are suitably interpreted. We consider particularly the group p2(X) associated with an abstract local group X. Let G be a groupoid.I We denote by it(g) and i,(g) the leftand rightidentities of elements g in G. We call g closed if iK(g) = ir(g) = i(g). We choose a fixed identity go in G and denote by Go the group of closed elements g such that i(g) = go. Let V be a set of closed elements of G together with their inverses and identities. For v E V, let G(v) consist of those elements g such that iz(g) go, i,(g) = iI(v). Let A = A (G, V) be the totality of formal expressions a = ( v1 * V1) ... (gn * Vn) where n _ 1, vi e V, gi e G(vi). We introduce an associative multiplication into A by juxtaposition and a rightand left-neutral element ao. We also introduce an inversion J (automorphism of period 2) by the rule