Closed Point Sets on a Manifold
Solomon Lefschetz · Annals of Mathematics · 1927
1. The object of this paper is to investigate certain topological simultaneous invariants of a manifold ulI and a closed subset G on iln4. These invariants are integers analogous to the Betti numbers and include them as special cases. They satisfy certain relations (formulas (4), (5), (6), of No. 8) which include practically all duality relations now known in Combinatorial Analysis Situs. When G is a complex Ck on Mw the invariants can be readily defined in terms of the complexes on Ck. For an arbitrary G however they may be defined in several modes depending upon the meaning attributed to the terms cycle on G, complex on G or with boundary on G. The desideratum to be fulfilled is to preserve the properties obtained in the case of a Ck, and it is fulfilled by the sequences of fractionary cycles and complexes by means of which we define the cycles and complexes in question. Similar sequences of ordinary cycles were recently investigated by Vietorist but appear to be inadequate for the purpose. However, in his paper (footnote 18) he refers to an invariant due to Brouwer which as we have shown (No. 27) coincides with our number R,,(G) (generalized Betti number or connection index). Alexandroff+ has also defined recently the Betti numbers of a closed set on an S, as limits of those of suitable approximating complexes. It is not known as yet however if this mode of approach leads to invariants satisfying our relations except in a very special case,? nor would it seem easy to prove. As a matter of fact the problem of the equivalence of the various modes of defining the invariants offers considerable interest. Our general method consists in treating first Ck by isolating it within a neighborhood whose residual is a manifold with boundary. From a few