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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.

MSC:
 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.
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References:
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