Concerning the Generators of Homotopy Groups of a Polyhedron

Let us collect before the proof some notations and elementary facts to be used later. Given a simplicial complex J, by the notations Hn(T) and Zn(J) will be meant respectively the nth homology group and the group of M-cycles of T, formed by using finite chains with integral coefficients. Let P be a connected polyhedron and 5 be the w-sphere zZt-o xi = 1 in Euclidean (w + 1)-space (n^2). Let „< =fasl and <p„' -dv." =<p,.". Now, let L be a simplicial decomposition of P and p* be a vertex of L. As usual,2 we are able to construct a simplicial complex L subject to the following conditions: (i) There is a one-to-one transformation / from the set of vertices of L to the set of all p9's where q is a vertex of L, and pq is a path in P from p* to q. (ii) Let g be the transformation which carries pq to q and let 6=gf. Then, any k + l mutually distinct vertices 50, git, • • • , §u of L span a ^-simplex of L if and only if 6(q0), 0(q~i), • • ■ , 0(g*) are the distinct vertices of a fe-simplex of L and f(qy) is the resultant of f{q,) multiplied by the path represented by the oriented segment from 6%) to 6%,).