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Poset fiber theorems. (English) Zbl 1086.55003

The theory of partially ordered sets is amenable to topological concepts since the chains in a poset \(P\) naturally form a simplicial complex \(\Delta(P)\), the order complex of the poset. A celebrated result in combinatorial topology, usually called “Quillen’s fiber lemma” [D. Quillen, Adv. Math. 28, 101–128 (1978; Zbl 0388.55007)] states that (the order complexes of) two posets \(P\) and \(Q\) are homotopy equivalent provided that there is a poset map between \(P\) and \(Q\) with contractible fibers.
The purpose of the paper under review is to establish a number of generalizations and variations of this result which can be subsumed under the following scheme: If there is a poset map from \(P\) to \(Q\) such that all its fibers \(f^{-1}(Q_{\leq q})\) are sufficiently well behaved, then certain properties of \(Q\) can be pulled back to \(P\); here \(Q_{\leq q}\) denotes the order ideal generated by \(q\in Q\). For instance, the following theorem is proved: Let \(f:P\to Q\) be a poset map such that for all \(q\in Q\) the fiber \(\Delta(f^{-1}(Q_{\leq q}))\) is \((\dim\Delta(f^{-1}(Q_{<q})))\)-connected. If \(\Delta(Q)\) is connected then \(\Delta(P)\) is homotopy equivalent to \(\Delta(Q)\vee \bigvee_{q\in Q} \Delta(f^{-1}(Q_{\leq q}))*\Delta(Q_{>q})\); here \(\vee\) is the wedge operation, \(*\) is the join, and \(Q_{>q}\) is the order filter of all elements in \(Q\) strictly greater than \(q\).
The proof uses diagrams of spaces as in [G. M. Ziegler and R. T. Živaljević, Math. Ann. 295, No. 3, 527–548 (1993; Zbl 0792.55002)]. Other results are concerned with simplicial homology, Cohen-Macaulayness, and equivariant versions.

MSC:

55P10 Homotopy equivalences in algebraic topology
05E25 Group actions on posets, etc. (MSC2000)
06A11 Algebraic aspects of posets
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[1] E. K. Babson, A combinatorial flag space, Ph. D. Thesis, MIT, 1993.
[2] Kenneth Baclawski, Cohen-Macaulay ordered sets, J. Algebra 63 (1980), no. 1, 226 – 258. · Zbl 0451.06004
[3] Anders Björner, Subspace arrangements, First European Congress of Mathematics, Vol. I (Paris, 1992) Progr. Math., vol. 119, Birkhäuser, Basel, 1994, pp. 321 – 370. · Zbl 0844.52008
[4] A. Björner, Topological methods, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1819 – 1872. · Zbl 0851.52016
[5] Anders Björner, Nerves, fibers and homotopy groups, J. Combin. Theory Ser. A 102 (2003), no. 1, 88 – 93. · Zbl 1030.55006
[6] Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler, Oriented matroids, Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge, 1993. Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler, Oriented matroids, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge, 1999. · Zbl 0773.52001
[7] Anders Björner and Michelle L. Wachs, Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1299 – 1327. · Zbl 0857.05102
[8] A. Björner, M.L. Wachs and V. Welker, On sequentially Cohen-Macaulay complexes and posets, in preparation. · Zbl 1247.05270
[9] A. Björner and V. Welker, Segre and Rees products of posets, with ring-theoretic applications, preprint, 2003 (http://arxiv.org/abs/math.CO/0312516). · Zbl 1062.05147
[10] Glen E. Bredon, Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer-Verlag, New York, 1993. · Zbl 0791.55001
[11] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. · Zbl 1044.55001
[12] Gene Freudenburg, Local slice constructions in \?[\?,\?,\?], Osaka J. Math. 34 (1997), no. 4, 757 – 767. · Zbl 0901.13018
[13] P.J. Hilton, An Introduction to Homotopy Theory, Cambridge Tracts in Mathematics and Mathematical Physics, 43, Cambridge University Press, 1953. · Zbl 0051.40302
[14] B. Mirzaii and W. van der Kallen, Homology stability for unitary groups, Doc. Math. 7 (2002), 143 – 166. · Zbl 0999.19005
[15] Jonathan Pakianathan and Ergün Yalçın, On commuting and noncommuting complexes, J. Algebra 236 (2001), no. 1, 396 – 418. · Zbl 0993.20016
[16] J. Scott Provan and Louis J. Billera, Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Oper. Res. 5 (1980), no. 4, 576 – 594. · Zbl 0457.52005
[17] Daniel Quillen, Homotopy properties of the poset of nontrivial \?-subgroups of a group, Adv. in Math. 28 (1978), no. 2, 101 – 128. · Zbl 0388.55007
[18] J. Shareshian, Some results on hypergraph matching complexes and \(p\)-group complexes of symmetric groups, preprint, 2000.
[19] J. Shareshian and M.L. Wachs, On the top homology of hypergraph matching complexes, in preparation. · Zbl 1188.20060
[20] Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. · Zbl 0838.13008
[21] Bernd Sturmfels and Günter M. Ziegler, Extension spaces of oriented matroids, Discrete Comput. Geom. 10 (1993), no. 1, 23 – 45. · Zbl 0783.52009
[22] Sheila Sundaram and Volkmar Welker, Group actions on arrangements of linear subspaces and applications to configuration spaces, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1389 – 1420. · Zbl 0945.05067
[23] J. Thévenaz and P. J. Webb, Homotopy equivalence of posets with a group action, J. Combin. Theory Ser. A 56 (1991), no. 2, 173 – 181. · Zbl 0752.05059
[24] Michelle L. Wachs, Whitney homology of semipure shellable posets, J. Algebraic Combin. 9 (1999), no. 2, 173 – 207. · Zbl 0922.06005
[25] M.L. Wachs, Topology of matching, chessboard, and general bounded degree graph complexes, Algebra Universalis, Special Issue in Memory of Gian-Carlo Rota, 49 (2003), 345-385. · Zbl 1092.05511
[26] M.L. Wachs, Bounded degree digraph and multigraph matching complexes, in preparation.
[27] M.L. Wachs, Poset fiber theorems and Dowling lattices, in preparation.
[28] James W. Walker, Homotopy type and Euler characteristic of partially ordered sets, European J. Combin. 2 (1981), no. 4, 373 – 384. · Zbl 0472.06004
[29] V. Welker, Partition Lattices, Group Actions on Arrangements and Combinatorics of Discriminants, Habilitationsschrift, Essen, 1996.
[30] Volkmar Welker, Günter M. Ziegler, and Rade T. Živaljević, Homotopy colimits — comparison lemmas for combinatorial applications, J. Reine Angew. Math. 509 (1999), 117 – 149. · Zbl 0995.55004
[31] P. J. Witbooi, Excisive triads and double mapping cylinders, Topology Appl. 95 (1999), no. 2, 169 – 172. · Zbl 0923.55012
[32] Günter M. Ziegler and Rade T. Živaljević, Homotopy types of subspace arrangements via diagrams of spaces, Math. Ann. 295 (1993), no. 3, 527 – 548. · Zbl 0792.55002
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