Order complexes of noncomplemented lattices are nonevasive

Kozlov, Dmitry

doi:10.1090/S0002-9939-98-05021-7

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