G-theorem Research Articles

Transversals and Colorings of Simplicial Spheres

Transversals and Colorings of Simplicial Spheres

Positive Plücker tree certificates for non-realizability

We introduce a new method for finding a non-realizability certificate of a simplicial sphere Σ: we exhibit a monomial combination of classical 3-term Plücker relations that yields a sum of products of determinants that are known to be positive in any realization of Σ; but their sum should vanish, contradiction. Using this technique, we prove for the first time the non-realizability of a balanced 2-neighborly 3-sphere constructed by Zheng, a family of highly neighborly centrally symmetric spheres constructed by Novik and Zheng, and several combinatorial prismatoids introduced by Criado and Santos. The method in fact works for orientable pseudo-manifolds, not just for spheres.

Large final polynomials from integer programming

We introduce a new method for finding a non-realizability certificate of a simplicial sphere Σ. It enables us to prove for the first time the non-realizability of a balanced 2-neighborly 3-sphere by Zheng, a family of highly neighborly centrally symmetric spheres by Novik and Zheng, and several combinatorial prismatoids introduced by Criado and Santos. The method, implemented in the polymake framework, uses integer programming to find a monomial combination of classical 3-term Plücker relations that must be positive in any realization of Σ; but since this combination should also vanish identically, the realization cannot exist. Previous approaches by Firsching, implemented using SCIP, and by Gouveia, Macchia and Wiebe, implemented using Singular and Macaulay2, are not able to process these examples.

Moment-angle manifolds and connected sums of simplicial spheres

Moment-angle manifolds and connected sums of simplicial spheres

Highly neighborly centrally symmetric spheres

In 1995, Jockusch constructed an infinite family of centrally symmetric 3-dimensional simplicial spheres that are cs-2-neighborly. Here we generalize his construction and show that for all d≥3 and n≥d+1, there exists a centrally symmetric d-dimensional simplicial sphere with 2n vertices that is cs-⌈d/2⌉-neighborly. This result combined with work of Adin and Stanley completely resolves the upper bound problem for centrally symmetric simplicial spheres.

Ear Decomposition and Balanced Neighborly Simplicial Manifolds

We find the first non-octahedral balanced 2-neighborly 3-sphere and the balanced 2-neighborly triangulation of the lens space $L(3,1)$. Each construction has 16 vertices. We show that there exists a balanced 3-neighborly non-spherical 5-manifold with 18 vertices. We also show that the rank-selected subcomplexes of a balanced simplicial sphere do not necessarily have an ear decomposition.

Combinatorial higher dimensional isoperimetry and divergence

In this paper we provide a framework for the study of isoperimetric problems in finitely generated groups, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions, one can restrict to simplicial spheres of particular shapes, called “round” and “unfolded”, provided that a bounded quasi-geodesic combing exists. We prove that the problem of estimating higher dimensional divergence as well can be restricted to round spheres. Applications of these results include a combinatorial analogy of the Federer–Fleming inequality for finitely generated groups.

Squeezed complexes

Given a shifted order ideal $U$, we associate to it a family of simplicial complexes $(\Delta_t(U))_{t\geq 0}$ that we call squeezed complexes. In a special case, our construction gives squeezed balls that were defined and used by Kalai to show that there are many more simplicial spheres than boundaries of simplicial polytopes. We study combinatorial and algebraic properties of squeezed complexes. In particular, we show that they are vertex decomposable and characterize when they have the weak or the strong Lefschetz property. Moreover, we define a new combinatorial invariant of pure simplicial complexes, called the singularity index, that can be interpreted as a measure of how far a given simplicial complex is from being a manifold. In the case of squeezed complexes $(\Delta_t(U))_{t\geq 0}$, the singularity index turns out to be strictly decreasing until it reaches (and stays) zero if $t$ grows.

Topological Prismatoids and Small Simplicial Spheres of Large Diameter

We introduce topological prismatoids, a combinatorial abstraction of the (geometric) prismatoids recently introduced by the second author to construct counter-examples to the Hirsch conjecture. We show that the “strong d-step Theorem” that allows to construct such large-diameter polytopes from “non-d-step” prismatoids still works at this combinatorial level. Then, using metaheuristic methods on the flip graph, we construct four combinatorially different non-d-step 4-dimensional topological prismatoids with 14 vertices. This implies the existence of 8-dimensional spheres with 18 vertices whose combinatorial diameter exceeds the Hirsch bound. These examples are smaller that the previously known examples by Mani and Walkup in 1980 (24 vertices, dimension 11). Our non-Hirsch spheres are shellable but we do not know whether they are realizable as polytopes.

Resolutions of co-letterplace ideals and generalizations of Bier spheres

We give the resolutions of co-letterplace ideals of posets in a completely explicit, very simple form. This generalizes and simplifies a number of linear resolutions in the literature, among them the Eliahou-Kervaire resolutions of strongly stable ideals generated in a single degree. Our method is based on a general result of K. Yanagawa using the canonical module of a Cohen-Macaulay Stanley-Reisner ring. We discuss in detail how the canonical module may effectively be computed, and from this derive directly the resolutions. A surprising consequence is that we obtain a large class of simplicial spheres comprehensively generalizing Bier spheres.

Wedge Operations and Torus Symmetries II

AbstractA fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. In a previous paper, the authors provided a new way to find all characteristic maps on a simplicial complex K(J) obtainable by a sequence of wedgings from K.The main idea was that characteristic maps on K theoretically determine all possible characteristic maps on a wedge of K.We further develop our previous work for classification of toric spaces. For a star-shaped simplicial sphere K of dimension n-1 with m vertices, the Picard number Pic(K) of K is m-n. We call K a seed if K cannot be obtained by wedgings. First, we show that for a fixed positive integer 𝓁, there are at most finitely many seeds of Picard 𝓁 number supporting characteristic maps. As a corollary, the conjecture proposed by V. V. Batyrev in is solved affirmatively.Secondly, we investigate a systematicmethod to find all characteristic maps on K(J) using combinatorial objects called (realizable) puzzles that only depend on a seed K. These two facts lead to a practical way to classify the toric spaces of fixed Picard number.

When does a discrete-time random walk in $\mathbb{R}^{n}$ absorb the origin into its convex hull?

We connect this question to a problem of estimating the probability that the image of certain random matrices does not intersect with a subset of the unit sphere Sn−1Sn−1. In this way, the case of a discretized Brownian motion is related to Gordon’s escape theorem dealing with standard Gaussian matrices. We show that for the random walk BMn(i),i∈NBMn(i),i∈N, the convex hull of the first CnCn steps (for a sufficiently large universal constant CC) contains the origin with probability close to one. Moreover, the approach allows us to prove that with high probability the π/2π/2-covering time of certain random walks on Sn−1Sn−1 is of order nn. For certain spherical simplices on Sn−1Sn−1, we prove an extension of Gordon’s theorem dealing with a broad class of random matrices; as an application, we show that CnCn steps are sufficient for the standard walk on ZnZn to absorb the origin into its convex hull with a high probability. Finally, we prove that the aforementioned bound is sharp in the following sense: for some universal constant c>1c>1, the convex hull of the nn-dimensional Brownian motion conv{BMn(t):t∈[1,cn]}conv⁡{BMn(t):t∈[1,cn]} does not contain the origin with probability close to one.

Realizability and inscribability for simplicial polytopes via nonlinear optimization

We show that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres. Thus we obtain a complete classification of neighborly polytopes of dimension $4$, $6$ and $7$ with $11$ vertices, of neighborly $5$-polytopes with $10$ vertices, as well as a complete classification of simplicial $3$-spheres with $10$ vertices into polytopal and non-polytopal spheres. Surprisingly many of the realizable polytopes are also inscribable.

SIMPLICIAL WEDGE COMPLEXES AND PROJECTIVE TORIC VARIETIES

Let K be a fan-like simplicial sphere of dimension n-1 such that its associated complete fan is strongly polytopal, and let v be a vertex of K. Let K(v) be the simplicial wedge complex obtained by applying the simplicial wedge operation to K at v, and let <TEX>$v_0$</TEX> and <TEX>$v_1$</TEX> denote two newly created vertices of K(v). In this paper, we show that there are infinitely many strongly polytopal fans <TEX>${\Sigma}$</TEX> over such K(v)'s, different from the canonical extensions, whose projected fans <TEX>${Proj_v}_i{\Sigma}$</TEX> (i = 0, 1) are also strongly polytopal. As a consequence, it can be also shown that there are infinitely many projective toric varieties over such K(v)'s such that toric varieties over the underlying projected complexes <TEX>$K_{{Proj_v}_i{\Sigma}}$</TEX> (i = 0, 1) are also projective.

Stellar Theory for Flag Complexes

Refining a basic result of Alexander, we show that two flag simplicial complexes are piecewise linearly homeomorphic if and only if they can be connected by a sequence of flag complexes, each obtained from the previous one by either an edge subdivision or its inverse. For flag spheres we pose new conjectures on their combinatorial structure forced by their face numbers, analogous to the extremal examples in the upper and lower bound theorems for simplicial spheres. Furthermore, we show that our algorithm to test the conjectures searches through the entire space of flag PL spheres of any given dimension.

M-vector analogue for the cd-index

A well-known conjecture of McMullen, proved by Billera, Lee and Stanley, describes the face numbers of simple polytopes. The necessary and sufficient condition is that the toric g-vector of the polytope is an M-vector, that is, the vector of dimensions of graded pieces of a standard graded algebra A. Recent work by Murai, Nevo and Yanagawa suggests a similar condition for the coefficients of the cd-index of a Gorenstein* poset P. The coefficients of the cd-index are conjectured to be the dimensions of graded pieces in a standard multigraded algebra A. We prove the conjecture for simplicial spheres and we give numerical evidence for general shellable spheres. In the simplicial case we construct the multi-graded algebra A explicitly using lattice paths.

Substitutions of polytopes and of simplicial complexes, and multigraded betti numbers

For a simplicial complex $K$ on $m$ vertices and simplicial complexes $K_1,\ldots ,K_m$, we introduce a new simplicial complex $K(K_1,\ldots ,K_m)$, called a substitution complex. This construction is a generalization of the iterated simplicial wedge studied by A. Bari, M. Bendersky, F. R. Cohen, and S. Gitler. In a number of cases it allows us to describe the combinatorics of generalized joins of polytopes $P(P_1,\ldots ,P_m)$, as introduced by G. Agnarsson. The substitution gives rise to an operad structure on the set of finite simplicial complexes in which a simplicial complex on $m$ vertices is considered as an $m$-ary operation. We prove the following main results: (1) the complex $K(K_1,\ldots ,K_m)$ is a simplicial sphere if and only if $K$ is a simplicial sphere and the $K_i$ are the boundaries of simplices, (2) the class of spherical nerve-complexes is closed under substitution, (3) multigraded betti numbers of $K(K_1,\ldots ,K_m)$ are expressed in terms of those of the original complexes $K, K_1,\ldots ,K_m$. We also describe connections between the obtained results and the known results of other authors.

From Flag Complexes to Banner Complexes

A notion of an $i$-banner simplicial complex is introduced. For various values of $i$, these complexes interpolate between the class of flag complexes and the class of all simplicial complexes. Examples of simplicial spheres of an arbitrary dimension that are $(i+1)$-banner but not $i$-banner are constructed. It is shown that several theorems for flag complexes have appropriate $i$-banner analogues. Among them are (1) the codimension-$(i+j-1)$ skeleton of an $i$-banner homology sphere $\Delta$ is $2(i+j)$-Cohen--Macaulay for all $0\leq j\leq \dim\Delta+1-i$, and (2) for every $i$-banner simplicial complex $\Delta$ there exists a balanced complex $\Gamma$ with the same number of vertices as $\Delta$ whose face numbers of dimension $i-1$ and higher coincide with those of $\Delta$.

Sliding Functor and Polarization Functor for Multigraded Modules

We define sliding functors, which are exact endofunctors of the category of multigraded modules over a polynomial ring. They preserve several invariants of modules, especially the (usual) depth and Stanley depth. In a similar way, we can also define the polarization functor. While this idea has appeared in papers of Bruns–Herzog and Sbarra, we give slightly different approach. Keeping these functors in mind, we treat simplicial spheres of Bier–Murai type.

LVMB manifolds and simplicial spheres

LVM and LVMB manifolds are a large family of non kähler manifolds. For instance, Hopf manifolds and Calabi-Eckmann manifolds can be seen as LVMB manifolds. The LVM manifolds have a natural action of a real torus and the quotient of this action is a polytope. This quotient allows us to relate closely LVM manifolds to the moment-angle manifolds studied by Buchstaber and Panov. Our aim is to generalize the polytope associated to a LVM manifold to the LVMB case and study the properties of this generalization. In particular, we show that the object we obtain belongs to a very large class of simplicial spheres. Moreover, we show that for every sphere belonging to this class, we can construct a LVMB manifold whose associated sphere is the given sphere. We use this latter result to show that many moment-angle complexes can be endowed with a complex structure (up to product with circles).