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  • 1. Adiprasito, Karim
    et al.
    Björner, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Goodarzi, Afshin
    Freie Universität, Germany.
    Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals2017Ingår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 19, nr 12, s. 3851-3865Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A numerical characterization is given of the h-triangles of sequentially Cohen-Macaulay simplicial complexes. This result determines the number of faces of various dimensions and codimensions that are possible in such a complex, generalizing the classical Macaulay-Stanley theorem to the nonpure case. Moreover, we characterize the possible Betti tables of componentwise linear ideals. A key tool in our investigation is a bijection between shifted multicomplexes of degree <= d and shifted pure. (d - 1)-dimensional simplicial complexes.

  • 2. Billera, L. J.
    et al.
    Björner, Anders.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Face numbers of polytopes and complexes2017Ingår i: Handbook of Discrete and Computational Geometry, Third Edition, CRC Press , 2017, s. 449-475Kapitel i bok, del av antologi (Övrigt vetenskapligt)
    Abstract [en]

    Geometric objects are often put together from simple pieces according to certain combinatorial rules. As such, they can be described as complexes with their constituent cells, which are usually polytopes and often simplices. Many constraints of a combinatorial and topological nature govern the incidence structure of cell complexes and are therefore relevant in the analysis of geometric objects. Since these incidence structures are in most cases too complicated to be well understood, it is worthwhile to focus on simpler invariants that still say something nontrivial about their combinatorial structure. The invariants to be discussed in this chapter are the f-vectors f = (f 0, f 1, …) $ f=(f_0, f_1, \dots) $, where f i $ f_i $ is the number of i-dimensional cells in the complex. 

  • 3.
    Björner, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A cell complex in number theory2011Ingår i: Advances in Applied Mathematics, ISSN 0196-8858, E-ISSN 1090-2074, Vol. 46, nr 1-4, s. 71-85Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let Delta(n) be the simplicial complex of squarefree positive integers less than or equal to n ordered by divisibility. It is known that the asymptotic rate of growth of its Euler characteristic (the Mertens function) is closely related to deep properties of the prime number system. In this paper we study the asymptotic behavior of the individual Betti numbers beta(k)(Delta(n)) and of their sum. We show that Delta(n) has the homotopy type of a wedge of spheres, and that as n -> infinity S beta(k)(Delta(n)) = 2n/pi(2) + O(n(theta)), for all theta > 17/54, Furthermore, for fixed k, beta k(Delta(n)) similar to n/2logn (log log n)(k)/k!. As a number-theoretic byproduct we obtain inequalities partial derivative(k)(sigma(odd)(k+1)(n)) infinity S beta k((Delta) over tilde (n)) = n/3 + O(n(theta)), for all theta > 22/27. 

  • 4.
    Björner, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A comparison theorem for f-vectors of simplicial polytopes2007Ingår i: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 3, nr 1, s. 347-356Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let f(i)(P) denote the number of i-dimensional faces of a convex polytope P. Furthermore, let S(n, d) and C(n, d) denote, respectively, the stacked and the cyclic d-dimensional polytopes on n vertices. Our main result is that for every simplicial d-polytope P, if f(r) (S (n(1), d)) <= f(r) (P) <= f(r) (C (n(2), d)) for some integers n(1), n(2) and r, then f(s) (S (n(1), d)) <= f(s) (P) <= f(s) (C (n(2), d)) for all s such that r < s. For r = 0 these inequalities are the well-known lower and upper bound theorems for simplicial polytopes. The result is implied by a certain comparison theorem for f-vectors, formulated in Section 4. Among its other consequences is a similar lower bound theorem for centrally-symmetric simplicial polytopes.

  • 5.
    Björner, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A {$q$}-analogue of the {FKG} inequality and some applications2011Ingår i: Combinatorica, ISSN 0209-9683, E-ISSN 1439-6912, Vol. 31, nr 2, s. 151-164Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let L be a finite distributive lattice and mu: L -> R(+) a log-supermodular function k: L -> R(+) let [GRAPHICS] We prove for any pair g, h: L -> R(+) of monotonely increasing functions, that E mu(g; q) . E mu(h; q) << E mu(1; q) . E mu(gh; q), where "<<" denotes coefficientwise inequality of real polynomials. The FKG inequality of Fortuin, Kasteleyn and Ginibre (1971) is the real number inequality obtained by specializing to q=1. The polynomial FKG inequality has applications to f-vectors of joins and intersections of simplicial complexes, to Betti numbers of intersections of Schubert varieties, and to correlation-type inequalities for a class of power series weighted by Young tableaux. This class contains series involving Plancherel measure for the symmetric groups and its poissonization.

  • 6.
    Björner, Anders
    KTH, Tidigare Institutioner                               , Matematik.
    Face numbers of Scarf complexes2000Ingår i: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 24, nr 3-Feb, s. 185-196Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let A be a (d + 1) x d real matrix whose row vectors positively span R-d and which is generic in the sense of Barany and Scarf [BS1]. Such a matrix determines a certain infinite d-dimensional simplicial complex Sigma, as described by Barany et al. [BHS]. The group Z(d) acts on Sigma with finitely many orbits. Let f(i) be the number of orbits of (i + 1)-simplices of Sigma. The sequence f = (f(0), f(1),..., f(d-1)) is the f-vector of a certain triangulated (d - 1)-ball T embedded in Sigma. When A has integer entries it is also, as shown by the work of Peeva and Sturmfels [PS], the sequence of Betti numbers of the minimal free resolution of k[x(1),...,x(d+1)]/I, where I is the lattice ideal determined by A. In this paper we study relations among the numbers f(i). It is shown that f(0), f(1),..., f([(d-3)/2]) determine the other numbers via linear relations, and that there are additional nonlinear relations. In more precise (and more technical) terms, our analysis shows that f is linearly determined by a certain M-sequence (g(0), g(1),..., g([(d-1)/2])). namely, the g-vector of the (d - 2)-sphere bounding T. Although T is in general not a cone over its boundary, it turns out that its f-vector behaves as if it were.

  • 7.
    Björner, Anders
    KTH, Tidigare Institutioner                               , Matematik.
    Nerves, fibers and homotopy groups2003Ingår i: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 102, nr 1, s. 88-93Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Two theorems are proved. One concerns coverings of a simplicial complex Delta by subcomplexes. It is shown that if every t-wise intersection of these subcomplexes is (k - t + 1)-connected, then for jless than or equal tok there are isomorphisms pi(j)(Delta) congruent to pi(j)(N) of homotopy groups of Delta and of the nerve X of the covering. The other concerns poset maps f : P --> Q. It is shown that if all fibers f(-1)(Q(less than or equal toq)) are k-connected, then f induces isomorphisms of homotopy groups pi(j)(P) congruent to pi(j)(Q), for all jless than or equal tok.

  • 8.
    Björner, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    NOTE: RANDOM-TO-FRONT SHUFFLES ON TREES2009Ingår i: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 14, s. 36-41Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A Markov chain is considered whose states are orderings of an underlying fixed tree and whose transitions are local "random-to-front" reorderings, driven by a probability distribution on subsets of the leaves. The eigenvalues of the transition matrix are determined using Brown's theory of random walk on semigroups.

  • 9.
    Björner, Anders.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Positive sum systems2015Ingår i: Springer INdAM Series, Springer International Publishing , 2015, s. 157-171Konferensbidrag (Refereegranskat)
    Abstract [en]

    Let x1, x2, …, xn be real numbers summing to zero, and let p+ be the family of all subsets J ⊆ [n]:={1,2,⋯n}such that (Formula presented). Subset families arising in this way are the objects of study here. We prove that the order complex of P+, viewed as a poset under set containment, triangulates a shellable ball whose f-vector does not depend on the choice of x, and whose h-polynomial is the classical Eulerian polynomial. Then we study various components of the flag f-vector of P+ and derive some inequalities satisfied by them. It has been conjectured by Manickam, Miklós and Singhi in 1986 that (Formula presented) is a lower bound for the number of k-element subsets in P+, unless n/k is too small. We discuss some related results that arise from applying the order complex and flag f-vector point of view. Some remarks at the end include brief discussions of related extensions and questions. For instance, we mention positive sum set systems arising in matroids whose elements are weighted by real numbers.

  • 10.
    Björner, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Random walks, arrangements, cell complexes, greedoids, and self-organizing libraries2008Ingår i: Building bridges: The Lovász Festschrift / [ed] Grötschel and G. O. H. Katona, Berlin: Springer Berlin/Heidelberg, 2008, s. 165-203Kapitel i bok, del av antologi (Refereegranskat)
    Abstract [en]

    The starting point is the known fact that some much-studied random walks on permutations, such as the Tsetlin library, arise from walks on real hyperplane arrangements. This paper explores similar walks on complex hyperplane arrangements. This is achieved by involving certain cell complexes naturally associated with the arrangement. In a particular case this leads to walks on libraries with several shelves.We also show that interval greedoids give rise to random walks belonging to the same general family. Members of this family of Markow chains, based on certain semigroups, have the property that all eigenvalues of the transition matrices are non-negative real and given by a simple combinatorial formula.Background material needed for understanding the walks is reviewed in rather great detail.

  • 11.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Ekedahl, Torsten
    On the shape of Bruhat intervals2009Ingår i: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 170, nr 2, s. 799-817Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let (W, S) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and let J subset of S. Let W-J denote the set of minimal coset representatives modulo the parabolic subgroup W-J. For w is an element of W-J, let f(i)(w,J) denote the number of elements of length i below w in Bruhat order on W-J (with notation simplified to f(i)(w) in the case when W-J = W). We show that 0 <= i < j <= l(w)-i implies f(i)(w,J) <= f(j)(w,J). Also, the case of equalities f(i)(w) = f(l(w)-i)(w) for i = 1,..., k is characterized in terms of vanishing of coefficients in the Kazhdan-Lusztig polynomial P-e,P-w (q). We show that if W is finite then the number sequence f(0)(w), f(1)(w),... f(l(w))(w) cannot grow too rapidly. Further, in the finite case, for any given k >= 1 and any w is an element of W of sufficiently great length (with respect to k), we show f(l(w)-k)(w) >= f(l(w)-k+1)(w) >= ... >= f(l(w))(w). The proofs rely mostly on properties of the cohomology of Kac-Moody Schubert varieties, such as the following result: if (X) over bar (w) is a Schubert variety of dimension d = l(w), and lambda = c(1) (L) is an element of H-2 ((X) over bar (w)) is the restriction to (X) over bar (w) of the Chem class of an ample line bundle, then (lambda(k)) . : Hd-k((X) over bar (w)) -> Hd+k((X) over bar (w)) is injective for all k >= 0.

  • 12.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Farley, J. D.
    Chain polynomials of distributive lattices are 75% unimodal2005Ingår i: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 12, nr 1Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    It is shown that the numbers c(i) of chains of length i in the proper part L\{0, 1} of a distributive lattice L of length l + 2 satisfy the inequalities c(0) <...< c([l/2]) and c([3l.4]) >... > c(l). This proves 75% of the inequalities implied by the Neggers unimodality conjecture.

  • 13.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Francesco, Brenti
    Dipartimento di Matematica, Seconda Universita di Roma.
    Combinatorics of {C}oxeter groups2005Bok (Övrigt vetenskapligt)
  • 14.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Goodarzi, A.
    On codimension one embedding of simplicial complexes2017Ingår i: A Journey through Discrete Mathematics: A Tribute to Jiri Matousek, Springer International Publishing , 2017, s. 207-219Kapitel i bok, del av antologi (Övrigt vetenskapligt)
    Abstract [en]

    We study d-dimensional simplicial complexes that are PL embeddable in Rd+1. It is shown that such a complex must satisfy a certain homological condition. The existence of this obstruction allows us to provide a systematic approach to deriving upper bounds for the number of top-dimensional faces of such complexes, particularly in low dimensions.

  • 15.
    Björner, Anders
    et al.
    KTH, Tidigare Institutioner, Matematik.
    Hultman, Axel
    KTH, Tidigare Institutioner, Matematik.
    A note on blockers in posets2004Ingår i: Annals of Combinatorics, ISSN 0218-0006, E-ISSN 0219-3094, Vol. 8, nr 2, s. 123-131Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The blocker A* of an antichain A in a finite poset P is the set of elements minimal with the property of having with each member of A a common predecessor. The following is done: (1) The posets P for which A** = A for all antichains are characterized.(2) The blocker A* of a symmetric antichain in the partition lattice is characterized.(3) Connections with the question of finding minimal size blocking sets for certain set families are discussed.

  • 16.
    Björner, Anders.
    et al.
    KTH, Tidigare Institutioner                               , Matematik.
    Lutz, F. H.
    Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincare homology 3-sphere2000Ingår i: Experimental Mathematics, ISSN 1058-6458, E-ISSN 1944-950X, Vol. 9, nr 2, s. 275-289Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present a computer program based on bistellar operations that provides a useful too I for the construction of simplicial manifolds with few vertices. As an example, we obtain a 16-vertex triangulation of the Poincare homology 3-sphere; we construct an infinite series of non-FL d-dimensional spheres with d + 13 vertices for d greater than or equal to 5; and we show that ifa d-manifold, with d greater than or equal to 5, admits any triangulation on n vertices, it admits a noncombinatorial triangulation on n + 12 vertices.

  • 17.
    Björner, Anders.
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Matoušek, J.
    Ziegler, G. M.
    Using brouwer’s fixed point theorem2017Ingår i: A Journey through Discrete Mathematics: A Tribute to Jiri Matousek, Springer International Publishing , 2017, s. 221-271Kapitel i bok, del av antologi (Övrigt vetenskapligt)
    Abstract [en]

    Brouwer’s fixed point theorem from 1911 is a basic result in topology- with a wealth of combinatorial and geometric consequences. In these lecture notes we present some of them, related to the game of HEX and to the piercing of multiple intervals. We also sketch stronger theorems, due to Oliver and others, and explain their applications to the fascinating (and still not fully solved) evasiveness problem. © Springer International Publishing AG 2017.

  • 18.
    Björner, Anders.
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Paffenholz, A.
    Sjöstrand, J.
    Ziegler, G. M.
    Bier spheres and posets2005Ingår i: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 34, nr 1, s. 71-86Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n -2)-spheres on 2n vertices, as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that cut across an ideal. Thus we arrive at a substantial generalization of Bier's construction: the Bier posets Bier(P, I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular generalized Bier spheres. In the case of Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P, I) inherit these properties. In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields many shellable spheres, most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres.

  • 19.
    Björner, Anders.
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Peeva, I.
    Sidman, J.
    Subspace arrangements defined by products of linear forms2005Ingår i: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 71, s. 273-288Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The vanishing ideal of an arrangement of linear subspaces in a vector space is considered, and the paper investigates when this ideal can be generated by products of linear forms. A combinatorial construction (blocker duality) is introduced which yields such generators in cases with a great deal of combinatorial structure, and examples are presented that inspired the work. A construction is given which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. Generic arrangements of points in P-2 and lines in P-3 are also considered.

  • 20.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Sagan, Bruce E.
    Rationality of the Mobius function of a composition poset2006Ingår i: Theoretical Computer Science, ISSN 0304-3975, E-ISSN 1879-2294, Vol. 359, nr 3-Jan, s. 282-298Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the zeta and Mobius functions of a partial order on integer compositions first studied by Bergeron, Bousquet-Melou, and Dulucq. The Mobius function of this poset was determined by Sagan and Vatter. We prove rationality of various formal power series in noncommuting variables whose coefficients are evaluations of the zeta function, zeta, and the Mobius function, mu. The proofs are either directly from the definitions or by constructing finite-state automata. We also obtain explicit expressions for generating functions obtained by specializing the variables to commutative ones. We reprove Sagan and Vatter's formula for it using this machinery. These results are closely related to those of Bjorner and Reutenauer about subword order, and we discuss a common generalization.

  • 21.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Stanley, Richard P.
    Massachusetts Institute of Technology.
    A combinatorial miscellany2010Bok (Refereegranskat)
  • 22.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Tancer, Martin
    Note: Combinatorial Alexander Duality-A Short and Elementary Proof2009Ingår i: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 42, nr 4, s. 586-593Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let X be a simplicial complex with ground set V. Define its Alexander dual as the simplicial complex X* = {sigma subset of V vertical bar V \ sigma is not an element of X}. The combinatorial Alexander duality states that the ith reduced homology group of X is isomorphic to the (vertical bar V vertical bar - i - 3) th reduced cohomology group of X* (over a given commutative ring R). We give a self-contained proof from first principles accessible to a nonexpert.

  • 23.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Vorwerk, Kathrin
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Connectivity of chamber graphs of buildings and related complexes2010Ingår i: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 31, nr 8, s. 2149-2160Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let Delta be a thick and locally finite building with the property that no edge of the associated Coxerer diagram has label "infinity". The chamber graph G(Delta), whose edges are the pairs of adjacent chambers in Delta is known to be q-regular for a certain number q = q(Delta). Our main result is that G(Delta) is q-connected in the sense of graph theory. In the language of building theory this means that every pair of chambers of Delta is connected by q pairwise disjoint galleries. Similar results are proved for the chamber graphs of Coxeter complexes and for order complexes of geometric lattices.

  • 24.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Vorwerk, Kathrin
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    On the connectivity of manifold graphs2015Ingår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 143, nr 10, s. 4123-4132Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant b_M for simplicial d-manifolds M taking values in the range 0 <= b_M <= d-1. The main result is that b_M influences connectivity in the following way: The graph of a d-dimensional simplicial compact manifold M is (2d - b_M)-connected. The parameter b_M has the property that b_M = 0 if the complex M is flag. Hence, our result interpolates between Barnette's theorem (1982) that all d-manifold graphs are (d+1)-connected and Athanasiadis' theorem (2011) that flag d-manifold graphs are 2d-connected. The definition of b_M involves the concept of banner triangulations of manifolds, a generalization of flag triangulations.

  • 25.
    Björner, Anders.
    et al.
    KTH, Tidigare Institutioner                               , Matematik.
    Wachs, M. L.
    Geometrically constructed bases for homology of partition lattices of types A, B and D2004Ingår i: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 11, nr 2Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the splitting basis for the homology of the partition lattice given in [20], thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let A be a central and essential hyperplane arrangement in R-d. Let R-1,..., R-k be the bounded regions of a generic hyperplane section of A. We show that there are induced polytopal cycles rho(Ri) in the homology of the proper part LA of the intersection lattice such that {rho(Ri)}(i=1,...,k) is a basis for (H) over tilde (d-2)((L) over bar (A)). This geometric method for constructing combinatorial homology bases is applied to the Coxeter arrangements of types A, B and D, and to some interpolating arrangements.

  • 26.
    Björner, Anders.
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Wachs, M. L.
    Welker, V.
    Poset fiber theorems2005Ingår i: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 357, nr 5, s. 1877-1899Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Suppose that f : P --> Q is a poset map whose fibers f(-1)(Qless than or equal to(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.

  • 27.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Wachs, Michelle
    Welker, Volkmar
    ON SEQUENTIALLY COHEN-MACAULAY COMPLEXES AND POSETS2009Ingår i: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 169, nr 1, s. 295-316Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The classes of sequentially Cohen-Macaulay and sequentially homotopy Cohen-Macaulay complexes and posets are studied. First, some different versions of the definitions are discussed and the homotopy type is determined. Second, it is shown how various constructions, such as join, product and rank-selection preserve these properties. Third, a characterization of sequential Cohen-Macaulayness for posets is given. Finally, in an appendix we outline connections with ring-theory and survey some uses of sequential Cohen-Macaulayness in commutative algebra.

  • 28.
    Björner, Anders.
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Welker, V.
    Segre and Rees products of posets, with ring-theoretic applications2005Ingår i: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 198, nr 3-Jan, s. 43-55Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We introduce (weighted) Segre and Rees products for posets and show that these constructions preserve the Cohen-Macaulay property over a field k and homotopically. As an application we show that the weighted Segre product of two affine semigroup rings that are Koszul is again Koszul. This result generalizes previous results by Crona on weighted Segre products of polynomial rings. We also give a new proof of the fact that the Rees ring of a Koszul affine semigroup ring is again Koszul. The paper ends with a list of some open problems in the area.

  • 29. De Concini, C.
    et al.
    Procesi, C.
    Björner, Anders.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Hyperplane arrangements and box splines2008Ingår i: The Michigan mathematical journal, ISSN 0026-2285, E-ISSN 1945-2365, Vol. 57, s. 201-225Artikel i tidskrift (Refereegranskat)
1 - 29 av 29
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