Connections between Differential Geometry and Topology
Sumner Byron Myers · Proceedings of the National Academy of Sciences · 1935
having no point of C on their interiors or boundaries, the second composed of the remaining cells of 2,,.The cells of the first class will form a sub-com- plex 2* of M,.The cells of the second class will not form a complex, since they may have cells of the first class on their boundaries; nevertheless, their duals will form a complex A. Moreover, the cells of the second class will determine a region Rn containing C. Now, there is no difficulty in extending Pontrjagin's relation of duality to the Betti Groups of 2* and A. Moreover, every cycle of S -C is homologous to a cycle of 2, for sufficiently large values of n and bounds in S -C if and only if the corresponding cycle of 2* bounds for sufficiently large values of n.On the other hand, the regions R. close down on the point set C as n increases indefinitely, in the sense that the intersection of all the RI's is precisely C. Thus, the proof of the relation of duality be- tween the Betti groups of C and S -C may be carried through as if the space S were of finite dimensionality.