Coronated polyhedra and coronated ANRs

TL;DR

This paper introduces the concept of coronated polyhedra, characterizes them via polyhedral resolutions, and explores their homological and homotopical properties, extending classical results to this new class of spaces.

Contribution

It defines coronated polyhedra, establishes their characterization through polyhedral resolutions, and analyzes their homology and homotopy invariants, extending shape theory results.

Findings

Coronated polyhedra are characterized by countable polyhedral resolutions.

Homology theories satisfying Milnor's axioms are invariants of strong shape for these spaces.

Certain homotopy exact sequences fail for coronated polyhedra, contrasting with classical cases.

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 $n=0$.