The Intersection of Chains on a Topological Manifold

Abstract

1. In previous papers, one of them in collaboration with S. Lefschetz,t the author has dealt with topological manifolds. A topological manifold, M1I,I, is a compact separable Hausdorff space (therefore metric) which has a complete set of neighborhoods each of which is a combinatorial n-cell (F. M., p. 393). The following properties are showln in F. M. and F. M. 2 to hold for ilI,: 1. the invariance of the homology characters; ? 2. the standard properties of the Kronecker Index of two chains on M1I. whose dimensions are p and l p; 3. the Poincare duality theorem. Property 1 was proved intrinsically, i. e. without imbedding M31,, in a Euclidean space of higher dimension and using the properties of the space residual to M,,. In 2, however, the imbedding space was used to prove that every non-bounding p-cycle oin M,, is cut by some (n p) -cycle oin MI with a Kronecker Index ? 1. From 2 follows 3. The present article makes no use of the imbedding theorem but defines intrinsically on ]II5 intersection cycles P17 (h = p + q n), for two chains, Cp and Cq, on ]II71 of dimensionality p and q, not meeting one another's boundaries; and proves intrinsically that the cycles thus obtained form a locally homologous family (L. T., p. 183) about the geometric intersection, G, (L. T., p. 182) of C. and Cq, thereby duplicating for 3M1, the salielnt theorem of the Lefschetz intersection theory for simplicial manifolds.