Net homotopy for compacta

Abstract

The important Hurewicz theory of homotopy groups(2) is applicable only to arc-wise connected spaces. If these groups are defined for a space which is not arc-wise connected, their significance is limited to the arc-component in which the base-point is chosen. One of the most interesting features of the homotopy group theory is its relation to homology groups. This relationship is expressed sharply in this theorem of Hurewicz: the n-dimensional integral homology group and the n-dimensional homotopy group of an arc-wise connected space are isomorphic if the homotopy groups of lower dimensions vanish. In this theorem the homology groups are, appropriately, the continuous or singular groups. More familiar homology theories for spaces are those of Cech or of Vietoris. The theorem of Hurewicz holds for these homology groups only if the space is assumed to have certain local properties such as local contractibility. The purpose of this paper is to define homotopy groups which are significant for non-arc-wise connected spaces, and which are suitably related to the Cech homology groups for spaces which are not locally connected. These new groups are defined in terms of nets(3). The theory of nets and of their homology groups is abstracted from the Cech theory. The nets which we consider here are derived from the nerves of finite coverings of,compact metric spaces. By limiting the discussion to compacta we can consider simultaneously the equivalent but more intuitive theory of neighborhood homotopy. Moreover we give examples of connected compacta which have satisfactory net-homotopy groups, yet which are beyond the range of the classical theory. The basic ingredients of any such theory are the concepts of mapping and of homotopy. In section II the standard concept of mapping is retained, but new definitions of homotopy are studied. One of these is based on nets, the other on neighborhoods. By means of machinery introduced in section I, these two homotopies are compared. In section III both basic concepts (mapping,