Coronated polyhedra and coronated ANRs

Abstract

Locally compact separable metrizable spaces are characterized among all metrizable spaces as those that admit a cofinal sequence $K_1\subset K_2\subset\cdots$ of compact subsets. Their \v{C}ech cohomology is well-understood due to Petkova's short exact sequence $0\to\lim^1 H^{n-1}(K_i)\to H^n(X)\to\lim H^n(K_i)\to 0$. We study a dual class of spaces. We call a metrizable space $X$ a "coronated polyhedron" if it contains a compactum $K$ such that $X\setminus K$ is a polyhedron. These include, apart from compacta and polyhedra, spaces such as the topologist's sine curve (or the Warsaw circle) and the comb (=comb-and-flea) space. The complement of every locally compact subset of $S^n$ is a coronated polyhedron. We prove that a metrizable space $X$ is a coronated polyhedron if and only if it admits a countable polyhedral resolution; or, equivalently, a sequential polyhedral resolution $\dots\to R_2\to R_1$. In the latter case, we establish a short exact sequence $0\to\lim^1 H_{n+1}(R_i)\to H_n(X)\to\lim H_n(R_i)\to 0$ for Steenrod-Sitnikov homology and also for any (extraordinary) homology theory satisfying Milnor's axioms of map excision and $\prod$-additivity. We also show that such homology theories are invariants of strong shape for coronated polyhedra. On the other hand, Quigley's short exact sequence $0\to\lim^1\pi_{n+1}(R_i)\to\pi_n(X)\to\lim\pi_n(R_i)\to 0$ for Steenrod homotopy of compacta fails for Steenrod-Sitnikov homotopy of coronated polyhedra, at least when $n=0$.