A Theorem in Algebraic Topology

Abstract

1. The terminology and notations used in this paper are taken from Lefschetz's Algebraic Topology.' This book will be referred to by the abbreviated title and will be used as the authority for such known results as are quoted in the course of the argument. The chief of these known results is the Alexander duality theorem, which is stated in a convenient form in L.A.T. p. 125. A difficulty which occurs throughout algebraic topology, and particularly in connection with the type of theorem proved in this paper, is the multiplicity of forms in which a general theorem appears. For example, the Alexander duality theorem appears in L.A.T. in five different guises, for general complexes on p. 125, for cornbinatorial manifolds on p. 204, for nets of complexes on p. 227, for topological spaces on p. 255, and for geometric manifolds on p. 304. In order to keep this paper as simple as possible, it has been necessary to state and prove the results in a single formulation only. The formulation which has been chosen is a purely combinatorial one; the extension of the results to general topological spaces could be carried out without excessive difficulty by the methods of L.A.T. The theorems of the present paper appeared in the first place as a by-product of an investigation in the theory of numbers.2 The application to the theory of numbers is, however, irrelevant here. From the topological point of view, the theorems are a kind of generalisation of the vell-known result of Lebesgue3 that, if an open set of points in n-dimensional Euclidean space is covered by a finite number of closed sets of sufficiently small diameter, then at least one group of (n + 1) of the closed sets has a common point. The theorems are also very closely connected with the well-known Phragmen-Brouwer theorems which relate the homology groups of the intersection of two sets of points to the homology groups of the two sets taken separately (see L.A.T., (22-3), p. 270); in fact, the theorems of this paper might be regarded simply as Phragmen-Brouwer theorems of a high degree of generality. In proving the theorems, however, no use will be made of the classical results of Lebesgue and Phragmen-Brouwer; it is simpler to build up the proofs independently from the start. 2. In this section the concepts to be used in the sequel are defined. The starting-point is a locally finite simple complex T, (see L.A.T., p. 91, and (47-1), p. 132). It is important that T, in the large should be allowved to be either a finite or an infinite complex. By the definition of a simple complex, T, is augmentable (see L.A.T. pp. 130-133). Since T, is required by the hypotheses of