Inverse Limits of Uniform Covering Maps

arXiv:0808.4119v2 [math.GN] 22 Dec 2009 INVERSE LIMITS OF UNIFORM COVERING MAPS B. LABUZ Abstract. In “Rips complexes and covers in the uniform category” the au- thors define, following James, covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given. This paper notes that the universal generalized uniform covering map is uniformly equivalent to the inverse limit of uniform covering maps and is therefore approximated by uniform covering maps. A characterization of gen- eralized uniform covering maps that are approximated by uniform covering maps is provided as well as a characterization of generalized uniform covering maps that are uniformly equivalent to the inverse limit of uniform covering maps. Inverse limits of group actions that induce generalized uniform covering maps are also treated. Contents 1. Introduction 1 2. Generalized uniform covering maps approximated by uniform covering maps 3 3. Inverse limits of generalized uniform covering maps 6 4. Inverse limits of regular generalized uniform covering maps 8 References 13 1. Introduction In [1], a theory of uniform covering maps and generalized uniform covering maps for uniform spaces is developed. In particular, it is shown that a locally uniform joinable chain connected space has a universal generalized uniform covering space and a path connected, uniformly path connected, and uniformly semilocally simply connected space has a universal uniform covering space. This paper points out that the universal generalized uniform covering map is uniformly equivalent to the inverse limit of uniform covering maps and is therefore approximated by uniform covering maps. Thus we consider when a generalized uniform covering map is approximated by uniform covering maps and when it is uniformly equivalent to the inverse limits of uniform covering maps. Inverse limits of uniform covering maps in general are also investigated as are the inverse limit of group actions that induce uniform covering maps. Date: December 22, 2009. 2000 Mathematics Subject Classification. Primary 55Q52; Secondary 55M10, 54E15. 1 2 B. LABUZ A good source for basic facts about uniform spaces is [4]. Let us recall some definitions and results from [1]. Given a function f : X →Y with X a uniform space, the function generates a uniform structure on Y if the family {f(E) : E is an entourage of X} forms a basis for a uniform structure on Y . If Y already has a uniform structure, the function generates that structure if and only if it is uniformly continuous and the image of every entourage of X is an entourage of Y . Given an entourage E of X, an E-chain in X is a finite sequence x1, . . . , xn such that (xi, xi+1) ∈E for each i ≤n. Inverses and concatenations of E-chains are defined in the obvious way. X is chain connected if for each entourage E of X and any x, y ∈X there is an E-chain starting at x and ending at y. A function f : X →Y from a uniform space X has chain lifting if for every entourage E of X there is an entourage F of X so that for any x ∈X, any f(F)-chain in Y starting at f(x) can be lifted to an E-chain in X starting at x. The function f has uniqueness of chain lifts if for every entourage E of X there is an entourage F ⊂E so that any two F-chains in X starting at the same point with identical images must be equal. Showing that f has unique chain lifting amounts to finding an entourage of X that is transverse to f. An entourage E0 is transverse to f if for any (x, y) ∈E0 with f(x) = f(y), we must have x = y. The function has unique chain lifting if it has both chain lifting and uniqueness of chain lifts. Define a function f : X →Y is a uniform covering map if it generates the uniform structure on Y and has unique chain lifting. Like in the setting of paths, we wish to have homotopies of chains. Homotopies between chains were successfully defined in [3]. The following is an equivalent definition from [1] that relies on homotopies already defined for paths. It utilizes Rips complexes which are a fundamental tool for studying chains in a uniform space. Given an entourage E of X the Rips complex R(X, E) is the subcomplex of the full complex over X whose simplices are finite E-bounded subsets of X. Any E-chain x1, . . . , xn determines a homotopy class of paths in R(X, E). Simply join successive terms xi, xi+1 by an edge path, i.e., a path along the edge joining xi and xi+1. Since only homotopy classes of paths will be considered any two such paths will be equivalent. Two E-chains starting at the same point x and ending at the same point y are E-homotopic relative endpoints if the corresponding paths in R(X, E) are homotopic relative endpoints. We wish to consider finer and finer chains in a space and therefore come to the concept of generalized paths. A generalized path is a collection of homotopy classes of chains α = {[αE]}E where E runs over all

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