Poset fiber theorems

Björner, Anders; Wachs, Michelle; Welker, Volkmar

doi:10.1090/S0002-9947-04-03496-8

Poset fiber theorems
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by Anders Björner, Michelle L. Wachs and Volkmar Welker PDF

Trans. Amer. Math. Soc. 357 (2005), 1877-1899 Request permission

Abstract:

Suppose that $f:P \to Q$ is a poset map whose fibers $f^{-1}(Q_{\le q})$ are sufficiently well connected. Our main result is a formula expressing the homotopy type of $P$ in terms of $Q$ and the fibers. Several fiber theorems from the literature (due to Babson, Baclawski and Quillen) are obtained as consequences or special cases. Homology, Cohen-Macaulay, and equivariant versions are given, and some applications are discussed.

References

E. K. Babson, A combinatorial flag space, Ph. D. Thesis, MIT, 1993.
Kenneth Baclawski, Cohen-Macaulay ordered sets, J. Algebra 63 (1980), no. 1, 226–258. MR 568572, DOI 10.1016/0021-8693(80)90033-2
Anders Björner, Subspace arrangements, First European Congress of Mathematics, Vol. I (Paris, 1992) Progr. Math., vol. 119, Birkhäuser, Basel, 1994, pp. 321–370. MR 1341828
A. Björner, Topological methods, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1819–1872. MR 1373690
Anders Björner, Nerves, fibers and homotopy groups, J. Combin. Theory Ser. A 102 (2003), no. 1, 88–93. MR 1970978, DOI 10.1016/S0097-3165(03)00015-3
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. MR 1226888
Anders Björner and Michelle L. Wachs, Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1299–1327. MR 1333388, DOI 10.1090/S0002-9947-96-01534-6
A. Björner, M.L. Wachs and V. Welker, On sequentially Cohen-Macaulay complexes and posets, in preparation.
A. Björner and V. Welker, Segre and Rees products of posets, with ring-theoretic applications, preprint, 2003 (http://arxiv.org/abs/math.CO/0312516).
Glen E. Bredon, Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer-Verlag, New York, 1993. MR 1224675, DOI 10.1007/978-1-4757-6848-0
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Gene Freudenburg, Local slice constructions in $k[X,Y,Z]$, Osaka J. Math. 34 (1997), no. 4, 757–767. MR 1618653
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
B. Mirzaii and W. van der Kallen, Homology stability for unitary groups, Doc. Math. 7 (2002), 143–166. MR 1911214
Jonathan Pakianathan and Ergün Yalçın, On commuting and noncommuting complexes, J. Algebra 236 (2001), no. 1, 396–418. MR 1808361, DOI 10.1006/jabr.1999.8501
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. MR 593648, DOI 10.1287/moor.5.4.576
Daniel Quillen, Homotopy properties of the poset of nontrivial $p$-subgroups of a group, Adv. in Math. 28 (1978), no. 2, 101–128. MR 493916, DOI 10.1016/0001-8708(78)90058-0
J. Shareshian, Some results on hypergraph matching complexes and $p$-group complexes of symmetric groups, preprint, 2000.
J. Shareshian and M.L. Wachs, On the top homology of hypergraph matching complexes, in preparation.
Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579
Bernd Sturmfels and Günter M. Ziegler, Extension spaces of oriented matroids, Discrete Comput. Geom. 10 (1993), no. 1, 23–45. MR 1215321, DOI 10.1007/BF02573961
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. MR 1340186, DOI 10.1090/S0002-9947-97-01565-1
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. MR 1092846, DOI 10.1016/0097-3165(91)90030-K
Michelle L. Wachs, Whitney homology of semipure shellable posets, J. Algebraic Combin. 9 (1999), no. 2, 173–207. MR 1679252, DOI 10.1023/A:1018694401498
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.
M.L. Wachs, Bounded degree digraph and multigraph matching complexes, in preparation.
M.L. Wachs, Poset fiber theorems and Dowling lattices, in preparation.
James W. Walker, Homotopy type and Euler characteristic of partially ordered sets, European J. Combin. 2 (1981), no. 4, 373–384. MR 638413, DOI 10.1016/S0195-6698(81)80045-5
V. Welker, Partition Lattices, Group Actions on Arrangements and Combinatorics of Discriminants, Habilitationsschrift, Essen, 1996.
Volkmar Welker, Günter M. Ziegler, and Rade T. Živaljević, Homotopy colimits—comparison lemmas for combinatorial applications, J. Reine Angew. Math. 509 (1999), 117–149. MR 1679169, DOI 10.1515/crll.1999.035
P. J. Witbooi, Excisive triads and double mapping cylinders, Topology Appl. 95 (1999), no. 2, 169–172. MR 1696447, DOI 10.1016/S0166-8641(97)00281-2
Günter M. Ziegler and Rade T. Živaljević, Homotopy types of subspace arrangements via diagrams of spaces, Math. Ann. 295 (1993), no. 3, 527–548. MR 1204836, DOI 10.1007/BF01444901