On Quillen's Theorem A for posets

JONATHAN ARIEL BARMAK

JOURNAL OF COMBINATORIAL THEORY SERIES A

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2011 vol. 118 p. 2445 - 2453

0097-3165

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 then 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.