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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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[1] Babson, E.; Kozlov, D.N., Complexes of graph homomorphisms, Israel J. math., 152, 285-312, (2006) · Zbl 1205.52009
[2] Baclawski, K., Cohen-Macaulay ordered sets, J. algebra, 63, 226-258, (1980) · Zbl 0451.06004
[3] J.A. Barmak, Algebraic topology of finite topological spaces and applications, PhD thesis, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 2009. · Zbl 1235.55001
[4] Barmak, J.A., Star clusters in independence complexes of graphs · Zbl 1288.57004
[5] Barmak, J.A.; Minian, E.G., Simple homotopy types and finite spaces, Adv. math., 218, 87-104, (2008) · Zbl 1146.57034
[6] Barmak, J.A.; Minian, E.G., One-point reductions of finite spaces, h-regular CW-complexes and collapsibility, Algebr. geom. topol., 8, 1763-1780, (2008) · Zbl 1227.55005
[7] J.A. Barmak, E.G. Minian, Strong homotopy types, nerves and collapses, Discrete Comput. Geom., doi:10.1007/s00454-011-9357-5, in press. · Zbl 1242.57019
[8] Björner, A., Topological methods, (), 1819-1872 · Zbl 0851.52016
[9] Björner, A., Nerves, fibers and homotopy groups, J. combin. theory ser. A, 102, 88-93, (2003) · Zbl 1030.55006
[10] Björner, A.; Wachs, M.L.; Welker, V., Poset fiber theorems, Trans. amer. math. soc., 357, 5, 1877-1899, (2005) · Zbl 1086.55003
[11] Cohen, M.M., Simplicial structures and transverse cellularity, Ann. of math., 85, 218-245, (1967) · Zbl 0147.42602
[12] Cohen, M.M., A course in simple homotopy theory, (1970), Springer-Verlag New York, Heidelberg, Berlin · Zbl 0261.57009
[13] Csorba, P., On the simple \(\mathbb{Z}_2\)-homotopy types of graph complexes and their simple \(\mathbb{Z}_2\)-universality, Canad. math. bull., 51, 535-544, (2008) · Zbl 1169.57022
[14] Dowker, C.H., Homology groups of relations, Ann. of math., 56, 84-95, (1952) · Zbl 0046.40402
[15] Glaser, L.C., Geometrical combinatorial topology I, (1970), Van Nostrand Reinhold New York · Zbl 0212.55603
[16] Hatcher, A., Higher simple homotopy theory, Ann. of math., 102, 101-137, (1975) · Zbl 0305.57009
[17] Hatcher, A., Algebraic topology, (2002), Cambridge University Press · Zbl 1044.55001
[18] Hog-Angeloni, C.; Metzler, W.; Sieradski, A.J., Two-dimensional homotopy and combinatorial group theory, London math. soc. lecture note ser., vol. 197, (1993), Cambridge University Press Cambridge, xii+412 pp · Zbl 0788.00031
[19] Kirby, R.C.; Siebenmann, L.C., Foundational essays on topological manifolds, smoothings, and triangulations, (1977), Princeton Univ. Press · Zbl 0361.57004
[20] Kozlov, D.N., Simple homotopy types of Hom-complexes, neighborhood complexes, lovász complexes, and atom crosscut complexes, Topology appl., 153, 2445-2454, (2006) · Zbl 1105.57021
[21] Lück, W., A basic introduction to surgery theory, (), 1-224 · Zbl 1045.57020
[22] McCord, M.C., Singular homology groups and homotopy groups of finite topological spaces, Duke math. J., 33, 465-474, (1966) · Zbl 0142.21503
[23] Milnor, J.W., Construction of universal bundles II, Ann. of math., 63, 430-436, (1956) · Zbl 0071.17401
[24] Quillen, D., Higher algebraic K-theory, I: higher K-theories, Lecture notes in math., vol. 341, (1973), pp. 85-147 · Zbl 0292.18004
[25] Quillen, D., Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. math., 28, 101-128, (1978) · Zbl 0388.55007
[26] Walker, J.W., Homotopy type and Euler characteristic of partially ordered sets, European J. combin., 2, 373-384, (1981) · Zbl 0472.06004
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