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On homological situation properties of complexes and closed sets

1943; American Mathematical Society; Volume: 54; Issue: 2 Linguagem: Inglês

10.1090/s0002-9947-1943-0008700-1

1088-6850

Paul Alexandroff,

Homotopy and Cohomology in Algebraic Topology

The purpose of this paper is to find and to study topological invariants which connect the homological properties of a space K with those of its closed subset A and of the open complement G = K\A, and thus help to characterize from the homological point of view the situation of A in K.In the case when K is simply connected (that is when the Betti groups of K are zero) the problem is solved by the duality theorems of Alexander, Pontrjagin, and Kolmogoroff, which determine the Betti groups of G through the Betti groups of A. In the other special case when K is a manifold the first duality theorems have been obtained by Pontrjagin and Lefschetz [9] in 1927-1928.All these results are special cases of the general theory to which the present paper is devoted and which gives the very elementary construction (of the so-called extension-and intersection-homomorphisms, section 11) dominating the whole variety of duality and other situation properties.The complete combinatorial theory is given in Chapter I for an arbitrary cell complex K and its closed subcomplex A. In Chapter III the same theory is generalized for locally bicompact normal spaces K and their closed sets A; this generalization is based on an approximation process developed in Chapter II.Chapter IV deals with manifolds and gives an elementary proof (combinatorial in character) of the Alexander-Pontrjagin duality in its most general form.All main results obtained are completely formulated in the first four sections of Chapter I (sections 11-14).Numerical consequences are given in section 16.Section 18 deals with the Phragmen-Brouwer problem, while in section 19 some quite elementary examples are given as illustration.The elementary known facts and notations used throughout this paper are systematized in the Introduction; its first part contains the group theoretical material, the second the needed information on complexes.Thus the present paper is practically independent of the previous literature on related subjects.There are only few references to my paper [l], and each of them may be read without reading the rest of that paper.