A proof of the invariance of certain constants of analysis situs

whose rôles are analogous to that played by connectivity in the theory of surfaces. The numbers P¿ and 77¿ are each greater by unity than the maximum number of ¿-dimensional cycles (cf. § 4 below) which may be traced in the manifold, but in calculating the numbers P,, certain conventions about sense are taken into account and the attention is confined to the sensed cycles. Now if a manifold be subdivided into a complex, or generalized polyhedron, then one plus the maximum number of independent ¿-dimensional cycles of the polyhedron (i. e., cycles made up of cells of the polyhedron) is also equal to Ri, or to Pi if the conventions about sense be adopted. This theorem, which is of considerable use in calculating the values of the numbers (P) and (R), has been proved by Poincaré on the assumption that the manifold, all the cycles of the manifold, and all the cells of the complex may be regarded as made up of a finite number of analytic pieces. But such an assumption opens the way to a theoretical objection in that the numbers (P) and (R) when calculated from the analytic cycles alone might conceivably fail to be topological invariants. To remove this objection, it would have to be shown that there never could exist a point-for-point continuous reciprocal correspondence between two manifolds possessing different numbers (P) and ( R ), even if the correspondence were not required to be analytic. In the following discussion, we shall take into account not only non-analytic cycles but also cycles possessing singularities of however complicated a nature.