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Inverse systems with simplicial bonding maps and cell structures. (English) Zbl 07430008

The main result of the paper is: given a topologically complete space \(X\) and a “defining family of closed covers” \(\mathcal{A}\) of \(X\) (i.e., it satisfies some local refinement condition and completeness condition), to construct an inverse system \({\mathbf F}_{\mathcal A} = (F_{\lambda}, \pi_{\lambda}^{\mu}, \Lambda)\) of simplicial complexes and simplicial bonding maps, and an inverse system \({\mathbf N}_{\mathcal A} = (N_{\lambda}, \pi_{\lambda}^{\mu}, \Lambda)\) such that the limit spaces \(F_{\infty} = N_{\infty}\), where \(N_{\lambda}\) is a subcomplex of \(F_{\lambda}\) which is almost the same as a cover, and to obtain a proper map \(\pi: N_{\infty} \to X\), and a continuous map \(p: X\to N_{\infty}\) so that \(\pi\circ p = {\mathrm {id}}_X\) and the two maps \(p\circ \pi\) and \({\mathrm {id}}_{N_{\infty}}\) are homotopic (so \(N_{\infty}\) is homotopy equivalent to \(X\)). The construction of the inverse system is based on a modification of the theorem by S. Mardešić [Fundam. Math. 114, 53–78 (1981; Zbl 0411.54019)] that every topological space \(X\) admits a polyhedral resolution.
The authors then show that if \(X\) is a compact Hausdorff space and \(\mathcal{A}\) is a family of locally finite, normal, closed covers of \(X\) satisfying the local refinement condition, then the inverse systems \({\mathbf F}_{\mathcal A}\) and \({\mathbf N}_{\mathcal A}\) are HPol-expansions of \(N_{\infty}\) and hence of \(X\), where an expansion is in the shape theoretical sense.
They also show that the restricted inverse system \({\mathbf F}^{(0)} = (F_{\lambda}^{(0)}, \pi_{\lambda}^{\mu}, \Lambda)\) is a cell structure in the sense of [W. Dębski and E. D. Tymchatyn, Topology Appl. 239, 293–307 (2018; Zbl 1390.54014)] representing a space canonically homeomorphic to \(X\).

MSC:

54C05 Continuous maps
54B35 Spectra in general topology
54D30 Compactness
54E15 Uniform structures and generalizations
78A70 Biological applications of optics and electromagnetic theory
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References:

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