Fine shape II: A Whitehead-type theorem

Abstract

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 $\pi_0$ and $\pi_1$ is a fine shape equivalence if and only if it induces isomorphisms on the $\pi_i$ (=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 $H_n(X)=0$ if and only if each compactum $K\subset X$ lies in a compactum $L\subset X$ such that the map $H_n(K)\to H_n(L)$ is trivial. A cornerstone result of the paper is purely algebraic: If a direct sequence of groups $\Gamma_0\to\Gamma_1\to\dots$ has trivial colimit, then it is trivial as an ind-group (i.e. each $\Gamma_i$ maps trivially to some $\Gamma_j$), as long as it has one of the following forms: $\bullet$ $\lim^1_i G_{i0}\to\lim^1_i G_{i1}\to\dots$, where the $G_{ij}$ are countable abelian groups; $\bullet$ $\lim_i G_{i0}\to\lim_i G_{i1}\to\dots$, where the $G_{ij}$ are finitely generated groups, which are either all abelian or satisfy the Mittag-Leffler condition for each $j$.