Strong bonding homology and cohomology

Abstract

In this paper we define new strong bonding homology and strong bonding cohomology groups of an arbitrary Tychonoff (i.e., completely regular) topological space X with coefficients in an Abelian group G, denoted by H ¯ m b d ( X ; G ) and H ¯ b d m ( X ; G ) , respectively, m ∈ Z . These groups connect strong homology groups with compact supports H ¯ m c ( X ; G ) and strong homology groups H ¯ m ( X ; G ) of X and the Alexander–Spanier cohomology groups H m ( X ; G ) and strong cohomology groups H ¯ m ( X ; G ) of X. We show that for paracompact Hausdorff spaces X, which are homologically locally connected ( hlc G ∞ ) with respect to strong homology with compact supports, H ¯ m b d ( X ; G ) are trivial. As a consequence, H ¯ m c ( X ; G ) ≈ H ¯ m ( X ; G ) , for m ∈ Z . Similarly, for strongly cohomologically locally connected ( sclc G ∞ ) spaces X with respect to strong cohomology, H ¯ b d m ( X ; G ) are trivial and thus, H m ( X ; G ) ≈ H ¯ m ( X ; G ) , for m ∈ Z . It is also shown that the paracompactness condition is indispensable. The subject of the paper is interesting because of close connections of local and global properties of X. It is proved that any resolution p ̲ : X → X ̲ = ( X λ , p λ λ ′ , Λ ) of X consisting of paracompact Hausdorff spaces X λ , which are hlc G ∞ and at the same time are homologically locally connected ( HLC G ∞ ) with respect to singular homology, i.e., X λ ∈ hlc G ∞ ∩ HLC G ∞ , λ ∈ Λ , determines the strong homology groups H ¯ m ( X ; G ) , m ∈ Z , even if it is not an ANR - resolution of X. Another consequence of this theory is the following nontrivial result: If ( X , A ) is a closed pair of hereditarily paracompact Hausdorff spaces such that X , A ∈ hlc G m (respectively, X , A ∈ sclc G m and X , A ∈ clc G m ), then the quotient space X / A ∈ hlc G m (respectively, X / A ∈ sclc G m and X / A ∈ clc m ), where hlc G m , shlc m and clc G m are the homological local m-connectedness with respect to strong homology with compact supports, the strongly cohomological local m-connectedness with respect to strong cohomology and the cohomological local m-connectedness with respect to Čech cohomology, respectively. Two examples of separable metric spaces X are exhibited such that H ¯ m c ( X ; G ) ≉ H ¯ m ( X ; G ) , which solves a problem from [S. Mardešić, Strong Shape and Homology, Springer-Verlag, Berlin, Heidelberg, New York, 2000, p. 457].