Fine shape II: A Whitehead-type theorem

TL;DR

This paper establishes a Whitehead-type theorem in fine shape theory for certain locally connected spaces, linking shape equivalences to homotopy group isomorphisms, and explores related algebraic and homological properties.

Contribution

It proves a new Whitehead theorem in fine shape theory for locally connected spaces with trivial homotopy groups, and provides algebraic results on direct sequences of groups with trivial colimits.

Findings

Whitehead theorem extended to fine shape for locally connected spaces

Homology triviality characterized by compactum inclusions

Algebraic result on trivial colimits of direct group sequences

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 $j$.