Lim colim versus colim lim. I

TL;DR

This paper investigates the non-commutativity of direct and inverse limits in a topological context, computing the 'commutator' and analyzing the properties of related homology and cohomology maps, especially for locally compact spaces.

Contribution

It provides new computations of the kernel and surjectivity of the natural map between Čech homology approximants and addresses an open problem for locally compact spaces.

Findings

The map $ au_X$ is surjective for locally compact spaces when $n=0$.

The kernel of the dual cohomology map is explicitly computed.

The paper introduces a new functor $ ext{lim}^1_{ ext{fg}}$ for analyzing cohomology.

Abstract

We study a model situation in which direct limit ($\text{colim}$) and inverse limit ($\lim$) do not commute, and offer some computations of their "commutator". The homology of a separable metrizable space $X$ has two well-known approximants: $qH_n(X)$ ("\v{C}ech homology") and $pH_n(X)$ ("\v{C}ech homology with compact supports"), which are not homology theories but are nevertheless interesting as they are $\lim\text{colim}$ and $\text{colim}\lim$ applied to homology of finite simplicial complexes. The homomorphism $\tau_X: pH_n(X)\to qH_n(X)$, which is a special case of the natural map $\text{colim}\lim\to\lim\text{colim}$, need not be either injective (P. S. Alexandrov, 1947) or surjective (E. F. Mishchenko, 1953), but its surjectivity for locally compact $X$ remains an open problem. In the case $n=0$ we obtain an affirmative solution of this problem. For locally compact $X$, the dual map in cohomology $pH^n(X)\to qH^n(X)$ is shown to be surjective and its kernel is computed, in terms of $\lim^1$ and a new functor $\lim^1_{\text{fg}}$. The original map $\tau_X$ is surjective and its kernel is computed when $X$ is a "coronated polyhedron", i.e. contains a compactum whose complement is a polyhedron.