Order complexes of noncomplemented lattices are nonevasive
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- by Dmitry N. Kozlov PDF
- Proc. Amer. Math. Soc. 126 (1998), 3461-3465 Request permission
Abstract:
We prove the following theorem: Let $\mathcal {L}$ be a finite lattice, $x\in \bar {\mathcal {L}}$. Assume $B$ is a set of elements of $\mathcal {L}$ which includes all complements of $x$ and is included in the set of all upper (lower) semicomplements of $x$. Then $\delta (\overline {\mathcal {L}\setminus B})$ is nonevasive, in particular collapsible. This generalizes results of several previous papers, where, in different generalities, it has been proved that the mentioned complex is contractible.References
- Kenneth Baclawski, Galois connections and the Leray spectral sequence, Advances in Math. 25 (1977), no. 3, 191–215. MR 470035, DOI 10.1016/0001-8708(77)90073-1
- Kenneth Baclawski and Anders Björner, Fixed points and complements in finite lattices, J. Combin. Theory Ser. A 30 (1981), no. 3, 335–338. MR 618539, DOI 10.1016/0097-3165(81)90030-3
- Anders Björner, Homotopy type of posets and lattice complementation, J. Combin. Theory Ser. A 30 (1981), no. 1, 90–100. MR 607041, DOI 10.1016/0097-3165(81)90042-X
- A. Björner, A general homotopy complementation formula, Discrete Math., to appear, preprint 1994, 7 pages.
- A. Björner, Topological methods, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1819–1872. MR 1373690
- Anders Björner and James W. Walker, A homotopy complementation formula for partially ordered sets, European J. Combin. 4 (1983), no. 1, 11–19. MR 694463, DOI 10.1016/S0195-6698(83)80003-1
- Marshall M. Cohen, A course in simple-homotopy theory, Graduate Texts in Mathematics, Vol. 10, Springer-Verlag, New York-Berlin, 1973. MR 0362320, DOI 10.1007/978-1-4684-9372-6
- Henry H. Crapo, The Möbius function of a lattice, J. Combinatorial Theory 1 (1966), 126–131. MR 193018, DOI 10.1016/S0021-9800(66)80009-1
- Jeff Kahn, Michael Saks, and Dean Sturtevant, A topological approach to evasiveness, Combinatorica 4 (1984), no. 4, 297–306. MR 779890, DOI 10.1007/BF02579140
- James R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR 755006
- Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
- Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112. MR 232393, DOI 10.1007/BF02684591
- Richard P. Stanley, Enumerative combinatorics. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1986. With a foreword by Gian-Carlo Rota. MR 847717, DOI 10.1007/978-1-4615-9763-6
Additional Information
- Dmitry N. Kozlov
- Affiliation: Department of Mathematics, 2-392, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
- Address at time of publication: Institute for Advanced Study, Olden Lane, Princeton, New Jersey 08540
- Email: kozlov@math.mit.edu, kozlov@math.kth.se
- Received by editor(s): February 25, 1997
- Communicated by: Jeffry Kahn
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 3461-3465
- MSC (1991): Primary 05E99, 06A09, 06B99
- DOI: https://doi.org/10.1090/S0002-9939-98-05021-7
- MathSciNet review: 1621965