Lim colim versus colim lim. I

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.