arXiv subjects

Satoshi Murai

Publications and source records attributed to Satoshi Murai.

At least 37 records · Page 2Linked to original sources

arXiv2017

Balanced subdivisions and flips on surfaces

In this paper, we show that two balanced triangulations of a closed surface are not necessary connected by a sequence of balanced stellar subdivisions and welds. This answers a question posed by Izmestiev, Klee and Novik. We also show that two balanced triangulations of a closed surface are connected by a sequence of three local operations, which we call the pentagon contraction, the balanced edge subdivision and the balanced edge weld. In addition, we prove that two balanced triangulations of the 2-sphere are connected by a sequence of pentagon contractions and their inverses if none of them are octahedral spheres.

math.CO↗

arXiv2016

Hessenberg varieties and hyperplane arrangements

Given a semisimple complex linear algebraic group $G$ and a lower ideal $I$ in positive roots of $G$, three objects arise: the ideal arrangement $\mathcal{A}_I$, the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$, and the regular semisimple Hessenberg variety $\mbox{Hess}(S,I)$. We show that a certain graded ring derived from the logarithmic derivation module of $\mathcal{A}_I$ is isomorphic to $H^*(\mbox{Hess}(N,I))$ and $H^*(\mbox{Hess}(S,I))^W$, the invariants in $H^*(\mbox{Hess}(S,I))$ under an action of the Weyl group $W$ of $G$. This isomorphism is shown for general Lie type, and generalizes Borel's celebrated theorem showing that the coinvariant algebra of $W$ is isomorphic to the cohomology ring of the flag variety $G/B$. This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map $H^*(G/B)\to H^*(\mbox{Hess}(N,I))$ announced by Dale Peterson and an affirmative answer to a conjecture of Sommers-Tymoczko are immediate consequences. We also give an explicit ring presentation of $H^*(\mbox{Hess}(N,I))$ in types $B$, $C$, and $G$. Such a presentation was already known in type $A$ or when $\mbox{Hess}(N,I)$ is the Peterson variety. Moreover, we find the volume polynomial of $\mbox{Hess}(N,I)$ and see that the hard Lefschetz property and the Hodge-Riemann relations hold for $\mbox{Hess}(N,I)$, despite the fact that it is a singular variety in general.

math.AG↗

arXiv2016

A generalized lower bound theorem for balanced manifolds

A simplicial complex of dimension $d-1$ is said to be balanced if its graph is $d$-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a generalization of their result to balanced triangulations of closed homology manifolds and balanced triangulations of orientable homology manifolds with boundary under an additional assumption that all proper links of these triangulations have the weak Lefschetz property. As a corollary, we show that if $Δ$ is an arbitrary balanced triangulation of any closed homology manifold of dimension $d-1 \geq 3$, then $2h_2(Δ) - (d-1)h_1(Δ) \geq 4{d \choose 2}(\tildeβ_1(Δ)-\tildeβ_0(Δ))$, thus verifying a conjecture by Klee and Novik. To prove these results we develop the theory of flag $h''$-vectors.

math.CO↗

arXiv2016

On stacked triangulated manifolds

We prove two results on stacked triangulated manifolds in this paper: (a) every stacked triangulation of a connected manifold with or without boundary is obtained from a simplex or the boundary of a simplex by certain combinatorial operations; (b) in dimension $d \geq 4$, if $Δ$ is a tight connected closed homology $d$-manifold whose $i$th homology vanishes for $1 < i < d-1$, then $Δ$ is a stacked triangulation of a manifold.These results give affirmative answers to questions posed by Novik and Swartz and by Effenberger.

math.GT↗

arXiv2016

Face numbers and the fundamental group

We resolve a conjecture of Kalai asserting that the $g_2$-number of any simplicial complex $Δ$ that represents a connected normal pseudomanifold of dimension $d\geq 3$ is at least as large as ${d+2 \choose 2}m(Δ)$, where $m(Δ)$ denotes the minimum number of generators of the fundamental group of $Δ$. Furthermore, we prove that a weaker bound, $h_2(Δ)\geq {d+1 \choose 2}m(Δ)$, applies to any $d$-dimensional pure simplicial poset $Δ$ all of whose faces of co-dimension $\geq 2$ have connected links. This generalizes a result of Klee. Finally, for a pure relative simplicial poset $Ψ$ all of whose vertex links satisfy Serre's condition $(S_r)$, we establish lower bounds on $h_1(Ψ),\ldots,h_r(Ψ)$ in terms of the $μ$-numbers introduced by Bagchi and Datta.

math.CO↗

arXiv2016

Lefschetz properties of balanced 3-polytopes

In this paper, we study Lefschetz properties of Artinian reductions of Stanley-Reisner rings of balanced simplicial $3$-polytopes. A $(d-1)$-dimensional simplicial complex is said to be balanced if its graph is $d$-colorable. If a simplicial complex is balanced, then its Stanley-Reisner ring has a special system of parameters induced by the coloring. We prove that the Artinian reduction of the Stanley-Reisner ring of a balanced simplicial $3$-polytope with respect to this special system of parameters has the strong Lefschetz property if the characteristic of the base field is not two or three. Moreover, we characterize $(2,1)$-balanced simplicial polytopes, i.e., polytopes with exactly one red vertex and two blue vertices in each facet, such that an analogous property holds. In fact, we show that this is the case if and only if the induced graph on the blue vertices satisfies a Laman-type combinatorial condition.

math.CO↗

arXiv2016

Face numbers of manifolds with boundary

We study face numbers of simplicial complexes that triangulate manifolds (or even normal pseudomanifolds) with boundary. Specifically, we establish a sharp lower bound on the number of interior edges of a simplicial normal pseudomanifold with boundary in terms of the number of interior vertices and relative Betti numbers. Moreover, for triangulations of manifolds with boundary all of whose vertex links have the weak Lefschetz property, we extend this result to sharp lower bounds on the number of higher-dimensional interior faces. Along the way we develop a version of Bagchi and Datta's $σ$- and $μ$-numbers for the case of relative simplicial complexes and prove stronger versions of the above statements with the Betti numbers replaced by the $μ$-numbers. Our results provide natural generalizations of known theorems and conjectures for closed manifolds and appear to be new even for the case of a ball.

math.CO↗

arXiv2016

A Duality in Buchsbaum rings and triangulated manifolds

Let $Δ$ be a triangulated homology ball whose boundary complex is $\partialΔ$. A result of Hochster asserts that the canonical module of the Stanley--Reisner ring of $Δ$, $\mathbb F[Δ]$, is isomorphic to the Stanley--Reisner module of the pair $(Δ, \partialΔ)$, $\mathbb F[Δ,\partial Δ]$. This result implies that an Artinian reduction of $\mathbb F[Δ,\partial Δ]$ is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of $\mathbb F[Δ]$. We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with boundary. We also provide an explicit algebraic interpretation of the $h"$-numbers of Buchsbaum complexes and use it to prove the monotonicity of $h"$-numbers for pairs of Buchsbaum complexes as well as the unimodality of $h"$-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold $g$-conjecture.

math.AC↗

arXiv2016

Uniformly Cohen-Macaulay simplicial complexes and almost Gorenstein* simplicial complexes

In this paper, we study simplicial complexes whose Stanley-Reisner rings are almost Gorenstein and have $a$-invariant zero. We call such a simplicial complex an almost Gorenstein* simplicial complex. To study the almost Gorenstein* property, we introduce a new class of simplicial complexes which we call uniformly Cohen-Macaulay simplicial complexes. A $d$-dimensional simplicial complex $Δ$ is said to be uniformly Cohen-Macaulay if it is Cohen-Macaulay and, for any facet $F$ of $Δ$, the simplicial complex $Δ\setminus\{F\}$ is Cohen-Macaulay of dimension $d$. We investigate fundamental algebraic, combinatorial and topological properties of these simplicial complexes, and show that almost Gorenstein* simplicial complexes must be uniformly Cohen-Macaulay. By using this fact, we show that every almost Gorenstein* simplicial complex can be decomposed into those of having one dimensional top homology. Also, we give a combinatorial criterion of the almost Gorenstein* property for simplicial complexes of dimension $\leq 2$.

math.AC↗

arXiv2015

Balanced generalized lower bound inequality for simplicial polytopes

A remarkable and important property of face numbers of simplicial polytopes is the generalized lower bound inequality, which says that the $h$-numbers of any simplicial polytope are unimodal. Recently, for balanced simplicial $d$-polytopes, that is simplicial $d$-polytopes whose underlying graphs are $d$-colorable, Klee and Novik proposed a balanced analogue of this inequality, that is stronger than just unimodality. The aim of this article is to prove this conjecture of Klee and Novik. For this, we also show a Lefschetz property for rank-selected subcomplexes of balanced simplicial polytopes and thereby obtain new inequalities for their $h$-numbers.

math.CO↗

arXiv2014

Collection of abstracts of the Workshop on Triangulations in Geometry and Topology at CG Week 2014 in Kyoto

This workshop about triangulations of manifolds in computational geometry and topology was held at the 2014 CG-Week in Kyoto, Japan. It focussed on computational and combinatorial questions regarding triangulations, with the goal of bringing together researchers working on various aspects of triangulations and of fostering a closer collaboration within the computational geometry and topology community. Triangulations are highly suitable for computations due to their clear combinatorial structure. As a consequence, they have been successfully employed in discrete algorithms to solve purely theoretical problems in a broad variety of mathematical research areas (knot theory, polytope theory, 2- and 3-manifold topology, geometry, and others). However, due to the large variety of applications, requirements vary from field to field and thus different types of triangulations, different tools, and different frameworks are used in different areas of research. This is why today closely related research areas are sometimes largely disjoint leaving potential reciprocal benefits unused. To address these potentials a workshop on Triangulations was held at Oberwolfach Research Institute in 2012. Since then many new collaborations between researchers of different mathematical communities have been established. Regarding the computational geometry community, the theory of manifolds continues to contribute to advances in more applied areas of the field. Many researchers are interested in fundamental mathematical research about triangulations and thus will benefit from a broad set of knowledge about different research areas using different techniques. We hope that this workshop brought together researchers from many different fields of computational geometry to have fruitful discussions which will lead to new interdisciplinary collaborations and solutions.

cs.CG↗

arXiv2014

Invariance of Pontrjagin classes for Bott manifolds

A Bott manifold is the total space of some iterated $\mathbb C P^1$-bundle over a point. We prove that any graded ring isomorphism between the cohomology rings of two Bott manifolds preserves their Pontrjagin classes. Moreover, we prove that such an isomorphism is induced from a diffeomorphism if the Bott manifolds are $\mathbb Z/2$-trivial, where a Bott manifold is called $\mathbb Z/2$-trivial if its cohomology ring with $\mathbb Z/2$-coefficient is isomorphic to that of a product of $\mathbb C P^1$'s.

math.AT↗

arXiv2013

Squarefree P-modules and the cd-index

In this paper, we introduce a new algebraic concept, which we call squarefree P-modules. This concept is inspired from Karu's proof of the non-negativity of the cd-indices of Gorenstein* posets, and supplies a way to study cd-indices from the viewpoint of commutative algebra. Indeed, by using the theory of squarefree P-modules, we give several new algebraic and combinatorial results on CW-posets. First, we define an analogue of the cd-index for any CW-poset and prove its non-negativity when a CW-poset is Cohen-Macaulay. This result proves that the h-vector of the barycentric subdivision of a Cohen-Macaulay regular CW-complex is unimodal. Second, we prove that the Stanley-Reisner ring of the barycentric subdivision of an odd dimensional Cohen-Macaulay polyhedral complex has the weak Lefschetz property. Third, we obtain sharp upper bounds of the cd-indices of Gorenstein* posets for a fixed rank generating function.

math.CO↗

arXiv2013

On ideals with the Rees property

A homogeneous ideal $I$ of a polynomial ring $S$ is said to have the Rees property if, for any homogeneous ideal $J \subset S $ which contains $I$, the number of generators of $J$ is smaller than or equal to that of $I$. A homogeneous ideal $I \subset S$ is said to be $\mathfrak m$-full if $\mathfrak mI:y=I$ for some $y \in \mathfrak m$, where $\mathfrak m$ is the graded maximal ideal of $S$. It was proved by one of the authors that $\mathfrak m$-full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals which have the Rees property but are not $\mathfrak m$-full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property.

math.AC↗

arXiv2013

On Hilbert functions of general intersections of ideals

Let I and J be homogeneous ideals in a standard graded polynomial ring. We study upper bounds of the Hilbert function of the intersection of I and g(J), where g is a general change of coordinates. Our main result gives a generalization of Green's hyperplane section theorem.

math.AC↗

arXiv2012

On r-stacked triangulated manifolds

The notion of r-stackedness for simplicial polytopes was introduced by McMullen and Walkup in 1971 as a generalization of stacked polytopes. In this paper, we define the r-stackedness for triangulated homology manifolds and study their basic properties. In addition, we find a new necessary condition for face vectors of triangulated manifolds when all the vertex links are polytopal.

math.CO↗

arXiv2012

Regularity bounds for binomial edge ideals

We show that the Castelnuovo-Mumford regularity of the binomial edge ideal of a graph is bounded below by the length of its longest induced path and bounded above by the number of its vertices.

math.AC↗

arXiv2012

On the generalized lower bound conjecture for polytopes and spheres

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If $P$ is a simplicial $d$-polytope then its $h$-vector $(h_0,h_1,...,h_d)$ satisfies $h_0 \leq h_1 \leq ... \leq h_{\lfloor \frac d 2 \rfloor}$. Moreover, if $h_{r-1}=h_r$ for some $r \leq \frac d 2$ then $P$ can be triangulated without introducing simplices of dimension $\leq d-r$. The first part of the conjecture was solved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. In this paper, we give a proof of the remaining part of the conjecture. In addition, we generalize this property to a certain class of simplicial spheres, namely those admitting the weak Lefschetz property.

math.CO↗