On Certain New Topological Invariants of a Manifold.

of the surfaces z, with the property that a curve y9 intersects a surface yj once or zero times, modulo T, according as i is equal to or different from j. In other words, the intersection of y9 with yj is determined by the coefficient of ysyj in the bilinear form ytyj (mod x) Let 7rt' denote the number of times, modulo r, that a curve y9 cuts the surface zj bounded by y. The numbers 7r-' are the components of a contravariant tensor of the second order, this time with respect to the variables yi. In the general case, there will be a tensor 'i corresponding to each value of the coefficients of torsion. The tensors 0ijk and Pijk are not derivable from the Betti numbers and coefficients of torsion of a manifold, as may be seen by comparing the two manifolds given in my previous note. By means of the tensors 7r8, it is even possible to distinguish between manifolds with the same group. There are, for instance, two distinct manifolds of group 5 such that the Heegaard diagram of each may be formed on an anchor ring, but such that the characteristic curve of one diagram is ab5, of the other a2b5. For the first manifold, the possible values of the tensor 7r" are 1 and 4, modulo 5, depending upon the choice of the fundamental surface yll; for the second manifold, the possible values are 2 and 3. Evidently, the above discussion may be extended from three to n dimensions.