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On Quillen’s theorem A for posets. (English) Zbl 1234.05237
Author’s abstract: A theorem of McCord of 1966 and Quillen’s Theorem A of 1973 provide sufficient conditions for a map between two posets to be a homotopy equivalence at the level of complexes. We give an alternative elementary proof of this result and we deduce also a stronger statement: under the hypotheses of the theorem, the map is not only a homotopy equivalence, but a simple homotopy equivalence. This leads to stronger formulations of the simplicial version of Quillen’s Theorem A, the Nerve Lemma and other known results. In particular we establish a conjecture of Kozlov on the simple homotopy type of the crosscut complex and we improve a well-known result of Cohen on contractible mappings.

05E45 Combinatorial aspects of simplicial complexes
05E99 Algebraic combinatorics
06A99 Ordered sets
55P10 Homotopy equivalences in algebraic topology
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
Full Text: DOI arXiv
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