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Publications (10 of 28) Show all publications
Adiprasito, K., Björner, A. & Goodarzi, A. (2017). Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals. Journal of the European Mathematical Society (Print), 19(12), 3851-3865
Open this publication in new window or tab >>Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals
2017 (English)In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 19, no 12, p. 3851-3865Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
European Mathematical Society Publishing House, 2017
Keywords
Simplicial complex, face numbers, Stanley-Reisner rings, sequential Cohen-Macaulayness, componentwise linear ideals
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-186130 (URN)10.4171/JEMS/755 (DOI)000415853100009 ()2-s2.0-85035041682 (Scopus ID)
Funder
Swedish Research Council, 2011-11677-88409-18
Note

QC 20171207

Available from: 2016-05-02 Created: 2016-05-02 Last updated: 2017-12-07Bibliographically approved
Björner, A. & Goodarzi, A. (2017). On codimension one embedding of simplicial complexes. In: A Journey through Discrete Mathematics: A Tribute to Jiri Matousek: (pp. 207-219). Springer International Publishing
Open this publication in new window or tab >>On codimension one embedding of simplicial complexes
2017 (English)In: A Journey through Discrete Mathematics: A Tribute to Jiri Matousek, Springer International Publishing , 2017, p. 207-219Chapter in book (Other academic)
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.

Place, publisher, year, edition, pages
Springer International Publishing, 2017
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-227838 (URN)10.1007/978-3-319-44479-6_9 (DOI)2-s2.0-85042427114 (Scopus ID)9783319444796 (ISBN)9783319444789 (ISBN)
Note

QC 20180515

Available from: 2018-05-15 Created: 2018-05-15 Last updated: 2018-10-19Bibliographically approved
Björner, A., Matoušek, J. & Ziegler, G. M. (2017). Using brouwer’s fixed point theorem. In: A Journey through Discrete Mathematics: A Tribute to Jiri Matousek: (pp. 221-271). Springer International Publishing
Open this publication in new window or tab >>Using brouwer’s fixed point theorem
2017 (English)In: A Journey through Discrete Mathematics: A Tribute to Jiri Matousek, Springer International Publishing , 2017, p. 221-271Chapter in book (Other academic)
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.

Place, publisher, year, edition, pages
Springer International Publishing, 2017
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-227847 (URN)10.1007/978-3-319-44479-6_10 (DOI)2-s2.0-85042428228 (Scopus ID)9783319444796 (ISBN)9783319444789 (ISBN)
Note

QC 20180514

Available from: 2018-05-14 Created: 2018-05-14 Last updated: 2018-05-14Bibliographically approved
Björner, A. & Vorwerk, K. (2015). On the connectivity of manifold graphs. Proceedings of the American Mathematical Society, 143(10), 4123-4132
Open this publication in new window or tab >>On the connectivity of manifold graphs
2015 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 143, no 10, p. 4123-4132Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2015
National Category
Discrete Mathematics
Identifiers
urn:nbn:se:kth:diva-140322 (URN)10.1090/proc/12415 (DOI)2-s2.0-84938252347 (Scopus ID)
Funder
Knut and Alice Wallenberg Foundation
Note

QC 20160602

Available from: 2014-01-21 Created: 2014-01-21 Last updated: 2017-12-06Bibliographically approved
Björner, A. (2015). Positive sum systems. In: Springer INdAM Series: (pp. 157-171). Springer International Publishing
Open this publication in new window or tab >>Positive sum systems
2015 (English)In: Springer INdAM Series, Springer International Publishing , 2015, p. 157-171Conference paper (Refereed)
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.

Place, publisher, year, edition, pages
Springer International Publishing, 2015
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-216898 (URN)10.1007/978-3-319-20155-9_27 (DOI)000410796100027 ()2-s2.0-85028593699 (Scopus ID)
Note

Export Date: 24 October 2017; Book Chapter; Correspondence Address: Björner, A.; Kungl. Tekniska Högskolan, Matematiska Inst.Sweden; email: bjorner@kth.se. QC 20171101

Available from: 2017-11-01 Created: 2017-11-01 Last updated: 2017-11-01Bibliographically approved
Björner, A. (2011). A cell complex in number theory. Advances in Applied Mathematics, 46(1-4), 71-85
Open this publication in new window or tab >>A cell complex in number theory
2011 (English)In: Advances in Applied Mathematics, ISSN 0196-8858, E-ISSN 1090-2074, Vol. 46, no 1-4, p. 71-85Article in journal (Refereed) Published
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. 

Keywords
Mertens function, Liouville function, Multicomplex, Cellular realization
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-33991 (URN)10.1016/j.aam.2010.09.007 (DOI)000290190600006 ()2-s2.0-79953722811 (Scopus ID)
Note
QC 20110523Available from: 2011-05-23 Created: 2011-05-23 Last updated: 2017-12-11Bibliographically approved
Björner, A. (2011). A {$q$}-analogue of the {FKG} inequality and some applications. Combinatorica, 31(2), 151-164
Open this publication in new window or tab >>A {$q$}-analogue of the {FKG} inequality and some applications
2011 (English)In: Combinatorica, ISSN 0209-9683, E-ISSN 1439-6912, Vol. 31, no 2, p. 151-164Article in journal (Refereed) Published
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.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-82460 (URN)10.1007/s00493-011-2644-1 (DOI)000293788700002 ()2-s2.0-80051510904 (Scopus ID)
Note
QC 20120220Available from: 2012-02-11 Created: 2012-02-11 Last updated: 2017-12-07Bibliographically approved
Björner, A. & Stanley, R. P. (2010). A combinatorial miscellany. Geneva: L'Enseignement Mathématique
Open this publication in new window or tab >>A combinatorial miscellany
2010 (English)Book (Refereed)
Place, publisher, year, edition, pages
Geneva: L'Enseignement Mathématique, 2010. p. 164
Series
Monographies de L'Enseignement Mathématique, ISSN 0425-0818 ; 42
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-82475 (URN)978-2-940264-09-4 (ISBN)
Note
QC 20120220Available from: 2012-02-11 Created: 2012-02-11 Last updated: 2012-02-20Bibliographically approved
Björner, A. & Vorwerk, K. (2010). Connectivity of chamber graphs of buildings and related complexes. European journal of combinatorics (Print), 31(8), 2149-2160
Open this publication in new window or tab >>Connectivity of chamber graphs of buildings and related complexes
2010 (English)In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 31, no 8, p. 2149-2160Article in journal (Refereed) Published
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.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-26645 (URN)10.1016/j.ejc.2010.06.005 (DOI)000282674700017 ()2-s2.0-77956182341 (Scopus ID)
Note
QC 20101203Available from: 2010-12-03 Created: 2010-11-26 Last updated: 2017-12-12Bibliographically approved
Björner, A. & Tancer, M. (2009). Note: Combinatorial Alexander Duality-A Short and Elementary Proof. Discrete & Computational Geometry, 42(4), 586-593
Open this publication in new window or tab >>Note: Combinatorial Alexander Duality-A Short and Elementary Proof
2009 (English)In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 42, no 4, p. 586-593Article in journal (Refereed) Published
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.

Keywords
Simplicial complex, Homology, Cohomology, Alexander dual, theorem
Identifiers
urn:nbn:se:kth:diva-18900 (URN)10.1007/s00454-008-9102-x (DOI)000271198900005 ()2-s2.0-70449517513 (Scopus ID)
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-7497-2764

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