Let us collect before the proof some notations and elementary facts to be used later. Given a simplicial complex J, by the notations Hn(J) and Zn(J) will be meant respectively the nth homology group and the group of n-cycles of J, formed by using finite chains with integral coefficients. Let P be a connected polyhedron and S be the n-sphere o0 x2 -1 in Euclidean (n+ 1)-space (n >2). Let 4 be a mapping: S--P. Then, given any s, s'ES, a path X.., in S from s to s' gives rise to a path q(X..') in P from +6(s) to q6(s').' We shall write 46.., for 4(X,). q?,,, is determined uniquely by s and s', since S is simply connected.' Clearly, 4,, =4s and O.. 4,'.', =4'... 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 sulbject to the following conditions: (i) There is a one-to-one transformation f from the set of vertices of L to the set of all p,'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 O=gf. Then, any k+1 mutually distinct vertices qo, ql, * , qk of L span a k-simplex of L if and only if O(qo), O(q1), * * * , O(qk) are the distinct vertices of a k-simplex of L and f(qi,) is the resultant of f(4,) multiplied by the path represented by the oriented segment from O(4j) to O(qi,).