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Transactions of the American Mathematical Society

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Poset fiber theorems

Authors: Anders Björner, Michelle L. Wachs and Volkmar Welker
Journal: Trans. Amer. Math. Soc. 357 (2005), 1877-1899
MSC (2000): Primary 05E25, 06A11, 55P10
Published electronically: July 22, 2004
MathSciNet review: 2115080
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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.

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  • 1. E. K. Babson, A combinatorial flag space, Ph. D. Thesis, MIT, 1993.
  • 2. K. Baclawski, Cohen-Macaulay ordered sets, J. Algebra 63 (1980), 226-258.MR 81m:06002
  • 3. A. Björner, Subspace arrangements, First European Congress of Mathematics, Paris 1992, A. Joseph et al. (Eds), Progress in Math., 119, Birkhäuser, 1994, pp. 321-370.MR 96h:52012
  • 4. A. Björner, Topological Methods, Handbook of Combinatorics, R. Graham, M. Grötschel and L. Lovász, (Eds), North-Holland, Amsterdam, 1995, pp. 1819-1872.MR 96m:52012
  • 5. A. Björner, Nerves, fibers and homotopy groups, J. Combin. Theory, Ser. A, 102 (2003), 88-93.MR 2004a:55018
  • 6. A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G.M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1993. Second edition, 1999.MR 95e:52023; MR 2000j:52016
  • 7. A. Björner and M.L. Wachs, Shellable nonpure complexes and posets, I, Trans. AMS 348 (1996), 1299-1327.MR 96i:06008
  • 8. A. Björner, M.L. Wachs and V. Welker, On sequentially Cohen-Macaulay complexes and posets, in preparation.
  • 9. A. Björner and V. Welker, Segre and Rees products of posets, with ring-theoretic applications, preprint, 2003 (
  • 10. G.E. Bredon, Topology and Geometry, Graduate Texts in Mathematics, 139, Springer-Verlag, New York-Heidelberg-Berlin, 1993.MR 94d:55001
  • 11. A. Hatcher, Algebraic Topology, Cambridge University Press, 2001.MR 2002k:55001
  • 12. J. Herzog, V. Reiner and V. Welker, The Koszul property in affine semigroup rings, Pacific J. Math. 186 (1998), 39-65.MR 99i:13010
  • 13. P.J. Hilton, An Introduction to Homotopy Theory, Cambridge Tracts in Mathematics and Mathematical Physics, 43, Cambridge University Press, 1953.MR 15:52c
  • 14. B. Mirzaii and W. van der Kallen, Homology stability for unitary groups, Documenta Math. 7 (2002), 143-166.MR 2003e:19007
  • 15. J. Pakianathan and E. Yalçin, On commuting and non-commuting complexes, J. Algebra 236 (2001), 396-418.MR 2002c:20036
  • 16. J.S. Provan and L.J. Billera, Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Oper. Res. 5 (1980), 576-594.MR 82c:52010
  • 17. D. Quillen, Homotopy properties of the poset of non-trivial $p$-subgroups of a group, Advances in Math. 28 (1978), 101-128MR 80k:20049
  • 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.
  • 20. R.P. Stanley, Combinatorics and Commutative Algebra, Second edition, Birkhäuser, Boston, 1995.MR 98h:05001
  • 21. B. Sturmfels and G.M. Ziegler, Extension spaces of oriented matroids, Discrete Comput. Geometry 10 (1993), 23-45.MR 94i:52015
  • 22. S. Sundaram and V. Welker, Group actions on arrangements and applications to configuration spaces, Trans. Amer. Math. Soc. 349 (1997), 1389-1420.MR 97h:52012
  • 23. J. Thévenaz and P.J. Webb, Homotopy equivalence of posets with a group action J. Combin. Theory, Ser. A 56 (1991), 173-181.MR 92k:20049
  • 24. M.L. Wachs, Whitney homology of semipure shellable posets, J. Algebraic Combinatorics 9 (1999), 173-207.MR 2000e:06004
  • 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.
  • 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. J.W. Walker, Homotopy type and Euler characteristic of partially ordered sets, Europ. J. Combinatorics 2 (1981), 373-384.MR 83g:06002
  • 29. V. Welker, Partition Lattices, Group Actions on Arrangements and Combinatorics of Discriminants, Habilitationsschrift, Essen, 1996.
  • 30. V. Welker, G.M. Ziegler and R.T. Zivaljevic, Homotopy colimits - comparison lemmas for combinatorial applications, J. Reine Angew. Mathematik (Crelles Journal) 509 (1999), 117-149.MR 2000b:55010
  • 31. P.J. Witbooi, Excisive triads and double mapping cylinders, Topology and its Applications 95(1999), 169-172. MR 2000d:55026
  • 32. G.M. Ziegler and R.T. Zivaljevic, Homotopy type of arrangements via diagrams of spaces, Math. Ann. 295 (1983), 527-548.MR 94c:55018

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Additional Information

Anders Björner
Affiliation: Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden

Michelle L. Wachs
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124

Volkmar Welker
Affiliation: Fachbereich Mathematik und Informatik, Universität Marburg, D-350 32 Marburg, Germany

Received by editor(s): July 25, 2002
Received by editor(s) in revised form: August 20, 2003
Published electronically: July 22, 2004
Additional Notes: The first author was supported by Göran Gustafsson Foundation for Research in Natural Sciences and Medicine, and by EC’s IHRP programme, grant HPRN-CT-2001-00272.
The second author was supported in part by National Science Foundation grants DMS 9701407 and DMS 0073760.
The third author was supported by Deutsche Forschungsgemeinschaft (DFG), and by EC’s IHRP programme, grant HPRN-CT-2001-00272.
Article copyright: © Copyright 2004 American Mathematical Society

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