On a generalization of Leray simplicial complexes

We study the question of realisability of iterated higher Whitehead products with a given form of nested brackets by simplicial complexes, using the notion of the moment-angle complex $Z_K$. Namely, we say that a simplicial complex $K$ realises an iterated higher Whitehead product $w$ if $w$ is a nontrivial element of $\pi_*(Z_K)$. The combinatorial approach to the question of realisability uses the operation of substitution of simplicial complexes: for any iterated higher Whitehead product $w$ we describe a simplicial complex $\partial\Delta_w$ that realises $w$. Furthermore, for a particular form of brackets inside $w$, we prove that $\partial\Delta_w$ is the smallest complex that realises $w$. We also give a combinatorial criterion for the nontriviality of the product $w$. In the proof of nontriviality we use the Hurewicz image of $w$ in the cellular chains of $Z_K$ and the description of the cohomology product of $Z_K$. The second approach is algebraic: we use the coalgebraic versions of the Koszul and Taylor complex for the face coalgebra of $K$ to describe the canonical cycles corresponding to iterated higher Whitehead products $w$. This gives another criterion for realisability of $w$.

The notion of apartitionable simplicial complex is extended to that of asignable partially ordered set. It is shown in a unified way that face lattices of shellable polytopal complexes, polyhedral cone fans, and oriented matroid polytopes, are all signable. Each of these classes, which are believed to be mutually incomparable, strictly contains the class of convex polytopes. A general sufficient condition, termedtotal signability, for a simplicial complex to satisfy McMullen's Upper Bound Theorem on the numbers of faces, is provided. The simplicial members of each of the three classes above are concluded to be partitionable and to satisfy the upper bound theorem. The computational complexity of face enumeration and of deciding partitionability is discussed. It is shown that under a suitable presentation, the face numbers of a signable simplicial complex can be efficiently computed. In particular, the face numbers of simplicial fans can be computed in polynomial time, extending the analogous statement for convex polytopes.

We show that questions concerning the topological B-model on a Calabi-Yau manifold in the Landau-Ginzburg phase can be rephrased in the language of commutative algebra.This yields interesting and very practical methods for analyzing the model.We demonstrate how the relevant "Ext" groups and superpotentials can be computed efficiently by computer algebra packages such as Macaulay.This picture leads us to conjecture a general description of D-branes in linear sigma models in terms of triangulated categories.Each phase of the linear sigma model is associated with a different presentation of the category of D-branes.

The goal of this paper is to generalize some of the existing toolkit of combinatorial algebraic topology in order to study the homology of abstract chain complexes. We define shellability of chain complexes in a similar way as for cell complexes and introduce the notion of regular chain complexes. In the case of chain complexes coming from simplicial complexes we recover the classical notions but, in contrast to the topological case, in the abstract setting shellings turn out to be a weaker homological invariant. In particular, we study special chain complexes, which are cones, and a class of regular chain complexes, which we call totally regular, for which we can obtain complete homological information if an additional condition is fulfilled.

The completely positive rank is an analogue of the topological covering dimension, defined for nuclear C ∗ C^* -algebras via completely positive approximations. These may be thought of as simplicial approximations of the algebra, which leads to the concept of piecewise homogeneous maps and a notion of noncommutative simplicial complexes. We introduce a technical variation of completely positive rank and show that the two theories coincide in many important cases. Furthermore, we analyze some of their properties; in particular we show that both theories behave nicely with respect to ideals and that they coincide with the covering dimension of the spectrum for certain continuous trace C ∗ C^* -algebras.

We consider a problem of calculating the loop space homology for so-called polyhedral products defined by an arbitrary simplicial complex K. A presentation of this homology algebra is obtained from the homology of the complements of diagonal subspace arrangements, which, in turn, is calculated using an infinite resolution of the exterior Stanley-Reisner algebra. We get an explicit presentation of the loop homology algebra for polyhedral products for classes of simplicial complexes such as flag complexes and the duals of sequentially Cohen-Macaulay complexes in terms of higher commutator products. We give a construction for the iteration of higher products and discuss the relationship between this problem and problems in commutative algebra.

The tensor functor called \alpha -induction produces a new unitary fusion category from a Frobenius algebra object, or a Q -system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of N to M arising from a subfactor N\subset M of finite index and finite depth, which gives a braided fusion category of endomorphisms of N . It is also understood in terms of Ocneanu’s graphical calculus. We study this \alpha -induction for bi-unitary connections, which provides a characterization of finite-dimensional nondegenerate commuting squares, and present certain 4 -tensors appearing in recent studies of 2 -dimensional topological order. We show that the resulting \alpha -induced bi-unitary connections are flat if we start with a commutative Frobenius algebra, or a local Q -system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented.

The emergence of Boij–Söderberg theory has given rise to new connections between combinatorics and commutative algebra. Herzog, Sharifan, and Varbaro recently showed that every Betti diagram of an ideal with a [math] -linear minimal resolution arises from that of the Stanley–Reisner ideal of a simplicial complex. In this paper, we extend their result for the special case of [math] -linear resolutions using purely combinatorial methods. Specifically, we show bijective correspondences between Betti diagrams of ideals with [math] -linear resolutions, threshold graphs, and anti-lecture-hall compositions. Moreover, we prove that any Betti diagram of a module with a [math] -linear resolution is realized by a direct sum of Stanley–Reisner rings associated to threshold graphs. Our key observation is that these objects are the lattice points in a normal reflexive lattice polytope.

Spheres arising from multicomplexes