2016/11/25 by Vesko Valov, Valov, Vesko
Mathematics · #54C55 (Secondary) #54F45 #55M10 #55M15 (Primary) #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1611.08347
openalex publication_date 2016/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
There are different definitions of homological dimension of metric compacta involving either Čech homology or exact (Steenrod) homology. In this paper we investigate the relation between these homological dimensions with respect to different groups. It is shown that all homological dimensions of a metric compactum X with respect to any field coincide provided X is homologically locally connected with respect to the singular homology up to dimension n=dim X. We also prove that any two-dimensional lc2 metric compactum X satisfies the equality dim(X times Y)=dim X+dim Y for any metric compactum Y. This improves the well known result of Kodama that every two-dimensional ANR is dimensionally full-valued. Actually, the condition X to be lc2 can be weaken to the existence at every point x a neighborhood V of x such that the inclusion homomorphism Hk(V;S1)→ Hk(X;S1) is trivial for all k=1,2.