Threshold graphs, shifted complexes, and graphical complexes

We consider a variety of connections between threshold graphs, shifted complexes, and simplicial complexes naturally formed from a graph. These graphical complexes include the independent set, neighborhood, and dominance complexes. We present a number of structural results and relations among them including new characterizations of the class of threshold graphs.     

Clique complexes and graph powers

We study the behaviour of clique complexes of graphs under the operation of taking graph powers. As an example we compute the clique complexes of powers of cycles, or, in other words, the independence complexes of circular complete graphs.     

Convex Hulls of f- and β-Vectors

In this paper we describe the convex hulls of the sets of f- and β-vectors of different classes of simplicial complexes on n vertices. These include flag complexes, order complexes of posets, matroid complexes, and general abstract simplicial complexes. As a result of this investigation, standard linear programming problems on these sets can be solved, including maximization of the Euler characteristics or of the sum of the Betti numbers. Received July 16, 1995, and in revised form May 1, 1996.     

Coordination properties of vic-isonitrosoimines in their copper (II) and palladium (II) complexes

Preparation and structural characterization of palladium (II) complexes of ligands III-V and copper (II) complexes of III are reported. The elemental analyses of the complexes show that the metal: ligand ratio is 1:2. The electrical conductance in acetone shows the non-electrolytic nature of the complexes. The diamagnetic character suggests a gross square-planar geometry for the palladium (II) complexes. Copper (II) complexes are paramagnetic with¼eff.～1·90 B.M. Spectral data suggest that in all the complexes the ligand coordinates to the metal (II) symmetrically through isonitroso-nitrogen and imine-nitrogen, forming a five membered chelate ring. Amine-exchange reactions of the complexes are discussed and compared on the basis of their structures.     

On f-Vectors and Relative Homology

We find strong necessary conditions on the f-vectors, Betti sequences, and relative Betti sequence of a pair of simplicial complexes. We also present an example showing that these conditions are not sufficient. If only the difference between two Betti sequences is specified, and not the individual Betti sequences, then the characterization is complete, and the characterization of all pairs of simplicial complexes matches the characterization of pairs of near-cones. Our necessary conditions rely upon a combinatorial decomposition of pairs of simplicial complexes that reflects the homology and relative homology of the complexes.     

Koszul Bicomplexes and Generalized Koszul Complexes in Projective Dimension One

We describe Koszul type complexes associated with a linear map from any module to a free module, and vice versa with a linear map from a free module to an arbitrary module, generalizing the classical Koszul complexes. Given a short complex of finite free modules, we assemble these complexes to what we call Koszul bicomplexes. They are used in order to investigate the homology of the Koszul complexes in projective dimension one. As in the case of the classical Koszul complexes, this homology turns out to be grade sensitive. In a special setup, we obtain necessary conditions for a map of free modules to be lengthened to a short complex of free modules.     

The complement of polyhedral product spaces and the dual simplicial complexes

In this paper, we define and prove basic properties of complement polyhedral product spaces, dual complexes and polyhedral join complexes. Then we compute the universal algebra of polyhedral join complexes under certain split conditions and the Alexander duality isomorphism on certain polyhedral product spaces.     

Noncommutative algebras associated to complexes and graphs

A new class of noncommutative algebras associated to complexes and graphs is introduced. Algebras associated to one-dimensional complexes are studied.     

Subword complexes, cluster complexes, and generalized multi-associahedra

In this paper, we use subword complexes to provide a uniform approach to finite-type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. For k=1, we show that this subword complex is isomorphic to the cluster complex of the given type. We show that multi-cluster complexes of types A and B coincide with known simplicial complexes, namely with the simplicial complexes of multi-triangulations and centrally symmetric multi-triangulations, respectively. Furthermore, we show that the multi-cluster complex is universal in the sense that every spherical subword complex can be realized as a link of a face of the multi-cluster complex.     

Complexes of directed trees and independence complexes

First we prove that certain complexes on directed acyclic graphs are shellable. Then we study independence complexes. Two theorems used for breaking and gluing such complexes are proved and applied to generalize the results by Kozlov.An interesting special case is anti-Rips complexes: a subset P of a metric space is the vertex set of the complex, and we include as a simplex each subset of P with no pair of points within distance r. For any finite subset P of R the homotopy type of the anti-Rips complex is determined.     

Matching Trees for Simplicial Complexes and Homotopy Type of Devoid Complexes of Graphs

We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes. We then exemplify their efficiency on some simplicial complexes, the devoid complexes of graphs, \(\mathcal {D}(G;\mathcal {F})\) whose faces are vertex subsets of G that induce \(\mathcal {F}\)-free subgraphs, where G is a multigraph and \(\mathcal {F}\) is a family of multigraphs. Additionally, we compute the homotopy type of dominance complexes of chordal graphs.     

Isodiametric and Isoperimetric Inequalities for Complexes and Groups

It is shown that D. Cohen's inequality bounding the isoperimetricfunction of a group by the double exponential of its isodiametricfunction is valid in the more general context of locally finitesimply connected complexes. It is shown that in this contextthis bound is ‘best possible’. Also studied aresecond-dimensional isoperimetric functions for groups and complexes.It is shown that the second-dimensional isoperimetric functionof a group is bounded by a recursive function. By a similarargument it is shown that the area distortion of a finitelypresented subgroup of a finitely presented group is recursive.Cohen's inequality is extended to second-dimensional isoperimetricand isodiametric functions of 2-connected simplicial complexes.     

Semi-tilting complexes

We introduce the notion of semi-tilting complexes, which is a small generalization of tilting complexes. Interesting examples include APR-semitilting complexes, etc. Note that non-trivial semi-tilting complexes exist for any non-semisimple non-local artin algebras, while tilting complexes may not. We extend interesting results in the tilting theory to semi-tilting complexes. As corollaries, we also obtain some new characterizations of tilting complexes.     

Directed Trees in a String,Real Polynomials with Triple Roots,and Chain Mails

This paper starts with an observation that two infinite series of simplicial complexes, which a priori do not seem to have anything to do with each other, have the same homotopy type. One series consists of the complexes of directed forests on a double directed string, while the other one consists of Shapiro–Welker models for the spaces of hyperbolic polynomials with a triple root. We explain this coincidence in the more general context by finding an explicit homotopy equivalence between complexes of directed forests on a double directed tree, and doubly disconnecting complexes of a tree.     

Extension dimension and quasi-finite CW-complexes

We extend the definition of quasi-finite complexes from countable complexes to arbitrary ones and provide a characterization of quasi-finite complexes in terms of L-invertible maps and dimensional properties of compactifications. Several results related to the class of quasi-finite complexes are established, such as completion of metrizable spaces, existence of universal spaces and a version of the factorization theorem. Furthermore, we define UV(L)-spaces in the realm of metrizable spaces and show that some properties of UV(n)-spaces and UV(n)-maps remain valid for UV(L)-spaces and UV(L)-maps, respectively.     

Depth for complexes, and intersection theorems

This paper introduces a new notion of depth for complexes; it agrees with the classical definition for modules, and coincides with earlier extensions to complexes, whenever those are defined. Techniques are developed leading to a quick proof of an extension of the Improved New Intersection Theorem (this uses Hochster's big Cohen-Macaulay modules), and also a generalization of the “depth formula” for tensor product of modules. Properties of depth for complexes are established, extending the usual properties of depth for modules. Received May 6, 1997; in final form December 3, 1997     

Gorenstein Flat and Cotorsion Dimensions of Unbounded Complexes

In this article, we define and study the Gorenstein flat dimension and Gorenstein cotorsion dimension for unbounded complexes over GF-closed rings by constructions of resolutions of unbounded complexes. The behavior of the dimensions under change of rings is investigated.     

Alperin's Weight Conjecture and Chain Complexes

It is shown that Alperin's weight conjecture is equivalent tothe existence of contractible chain complexes whose entrieshave the right dimension coming from some of the alternatingsum formulations. It is conjectured that for the other formulationsand for Dade's ordinary conjecture, there also exist such contractiblechain complexes.     

2-Groupoid Enrichments in Homotopy Theory and Algebra

The use of groupoid enrichments in abstract homotopy theory is well known and classical. Recently enrichments by higher-dimensional groupoids have been considered. Here we will describe enrichment by 2-groupoids with respect to the Gray tensor product and will examine several examples (2-groupoids, 2-crossed complexes, chain complexes, etc.) from an elementary view-point. The enrichment of the category of chain complexes is examined in detail and questions of the existence of analogues of classical constructions (categories over B, under A, etc.) are explored.