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One-point reductions of finite spaces, \(h\)-regular CW-complexes and collapsibility. (English) Zbl 1227.55005

Summary: We investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of \(h\)-regular CW-complex, generalizing the concept of regular CW-complex, and prove that the \(h\)-regular CW-complexes, which are a sort of combinatorial-up-to-homotopy objects, are modeled (up to homotopy) by their associated finite spaces. This is accomplished by generalizing a classical result of McCord on simplicial complexes.

MSC:

55P15 Classification of homotopy type
55U05 Abstract complexes in algebraic topology
57Q05 General topology of complexes
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
06A06 Partial orders, general
52B70 Polyhedral manifolds
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