2022/11/20 by Sergey A. Melikhov, Melikhov, Sergey A.
Mathematics · #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2211.11102
openalex publication_date 2022/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove an "abelian, locally compact" Whitehead theorem in fine shape: A fine shape morphism between locally connected finite-dimensional locally compact separable metrizable spaces with trivial π0 and π1 is a fine shape equivalence if and only if it induces isomorphisms on the πi (=the Steenrod-Sitnikov homotopy groups). We show by an example that the hypothesis of local connectedness cannot be dropped (even though it can be dropped in the compact case). As a byproduct, we also show that for a locally compact separable metrizable space X, the Steenrod-Sitnikov homology Hn(X)=0 if and only if each compactum K⊂ X lies in a compactum L⊂ X such that the map Hn(K)→ Hn(L) is trivial. A cornerstone result of the paper is purely algebraic: If a direct sequence of groups Γ0→Γ1→… has trivial colimit, then it is trivial as an ind-group (i.e. each Γi maps trivially to some Γj), as long as it has one of the following forms: \bullet lim1i Gi0→lim1i Gi1→…, where the Gij are countable abelian groups; \bullet limi Gi0→limi Gi1→…, where the Gij are finitely generated groups, which are either all abelian or satisfy the Mittag-Leffler condition for each j.