Аннотация:
Locally compact separable metrizable spaces are characterized among all metrizable spaces as those that admit a
cofinal sequence $K_1\subset K_2\subset \ldots$ of compact subsets. Their Čech cohomology is well-understood due to the following short exact sequence, which first appeared in a paper by S.V.Petkova, a student of E.G.Sklyarenko
[3; Proposition 4]: $0 \to \mathrm{lim}^1 H^{n-1}(K_i) \to H^n(X) \to \mathrm{lim} H^n(K_i) \to 0$.
(Apparently this paper [3] was the main part of her dissertation written under Sklyarenko's guidance. Its main results are improved versions of the main results of Sklyarenko's paper [2].) It should be noted that Petkova's short exact sequence was later rediscovered by W.Massey [4; Theorem 4.22]. In fact, it was almost discovered by A.Grothendieck in 1957 [1; Proposition 3.10.2], but not quite, because he was not aware of the $\mathrm{lim}^1$ functor at that time.
We study a dual class of spaces, which we show to satisfy a dual short exact sequence in homology [5]. We call a metrizable space $X$ a "coronated polyhedron" if it contains a compactum $K$ such that $X-K$ 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 $\ldots \to R_2 \to R_1$. ("Resolution" is in the sense of Mardešić.) In the latter case, we establish a short exact sequence $0 \to \mathrm{lim}^1 H_{n+1}(R_i) \to H_n(X) \to \mathrm{lim} H_n(R_i) \to 0$ for Steenrod-Sitnikov homology and also for any (possibly extraordinary) homology theory satisfying Milnor's axioms of map excision and $\prod$-additivity. On the other hand, Quigley's short exact sequence $0 \to \mathrm{lim}^1 \pi_{n+1}(R_i) \to \pi_n(X) \to \mathrm{lim} \pi_n(R_i) \to 0$ for Steenrod homotopy of compacta fails for Steenrod-Sitnikov homotopy of coronated polyhedra, at least when $n=0$.
[1] A. Grothendieck, Sur quelques points d’algèbre homologique, Tohoku
Math. J. 9 (1957), 119–221; Russian transl., А. Гротендик, О некоторых вопросах
гомологической алгебры, ИЛ, Москва, 1961.
[2] E. G. Sklyarenko, Uniqueness theorems in homology theory, Mat.
Sb. 85 (1971), 201–223; English transl., Math. USSR–Sb. 14 (1971),
199–218.
[3] S. V. Petkova, On the axioms of homology theory, Mat. Sbornik 90
(1973), 607–624; English transl., Math. USSR-Sb. 90 (1974), 597–614.
[4] W. S. Massey, Homology and Cohomology Theory, Marcel Dekker, New
York, 1978; Russian transl., У. Масси, Теория гомологий и когомологий,
Мир, Москва, 1981.
[5] S. A. Melikhov, Coronated polyhedra and coronated ANRs, arXiv:2211.09951.
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