arXiv subjects

Kiyoshi Igusa

Publications and source records attributed to Kiyoshi Igusa.

At least 37 records · Page 2Linked to original sources

arXiv2023

On Clusters and Exceptional Sets in Types $\mathbb{A}$ and $\tilde{\mathbb{A}}$

In this paper we first study clusters in type $\tilde{\mathbb{A}}$ by collecting them into a finite number of infinite families given by Dehn twists of their corresponding triangulations, and show that these families are counted by the Catalan numbers. We also highlight the similarities and differences between the annuli diagrams used to study clusters and those used to study exceptional sets in type $\tilde{\mathbb{A}}$. We then focus on exceptional collections (sets) of modules over path algebras of quivers by first showing that the notion of relative projectivity in exceptional sets is well defined. We finish by counting the number of exceptional sets of representations of type $\mathbb{A}$ quivers with straight orientation and using this to count the number of families of exceptional sets of type $\tilde{\mathbb{A}}$ with straight orientation.

math.RT↗

arXiv2023

Scaled Homology and Topological Entropy

In this paper, we build up a scaled homology theory, $lc$-homology, for metric spaces such that every metric space can be visually regarded as "locally contractible" with this newly-built homology. We check that $lc$-homology satisfies all Eilenberg-Steenrod axioms except exactness axiom whereas its corresponding $lc$-cohomology satisfies all axioms for cohomology. This homology can relax the smooth manifold restrictions on the compact metric space such that the entropy conjecture will hold for the first $lc$-homology group.

math.AT↗

arXiv2023

Pairwise Compatibility for 2-Simple Minded Collections

In $τ$-tilting theory, it is often difficult to determine when a set of bricks forms a 2-simple minded collection. The aim of this paper is to determine when a set of bricks is contained in a 2-simple minded collection for a $τ$-tilting finite algebra. We begin by extending the definition of mutation from 2-simple minded collections to more general sets of bricks (which we call semibrick pairs). This gives us an algorithm to check if a semibrick pair is contained in a 2-simple minded collection. We then use this algorithm to show that the 2-simple minded collections of a $τ$-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) are given by pairwise compatibility conditions if and only if every vertex in the corresponding quiver has degree at most 2. As an application, we show that the classifying space of the $τ$-cluster morphism category of a $τ$-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) is an Eilenberg- MacLane space if every vertex in the corresponding quiver has degree at most 2.

math.RT↗

arXiv2023

Linearity of stability conditions

For modules over an artin algebra a linear stability condition is given by a "central charge" and a nonlinear stability condition is given by the wall-crossing sequence of a "green path". Finite Harder-Narasimhan stratifications of the module category, maximal forward hom-orthogonal sequences and maximal green sequences, defined using Fomin-Zelevinsky quiver mutation are shown to be equivalent to finite nonlinear stability conditions when the algebra is hereditary. This is the first of a series of three papers whose purpose is to determine all maximal green sequences of maximal length for quivers of affine type A and determine which are linear. See [1].

math.RT↗

arXiv2023

Continuous Stability Conditions of Type A and Measured Laminations of the Hyperbolic Plane

We introduce stability conditions (in the sense of King) for representable modules of continuous quivers of type A along with a special criteria called the four point condition. The stability conditions are defined using a generalization of delta functions, called half-delta functions. We show that for a continuous quiver of type A with finitely many sinks and sources, the stability conditions satisfying the four point condition are in bijection with measured laminations of the hyperbolic plane. Along the way, we extend an earlier result by the first author and Todorov regarding continuous cluster categories for linear continuous quivers of type A and laminations of the hyperbolic plane to all continuous quivers of type A with finitely many sinks and sources. We also give a formula for the continuous cluster character.

math.RT↗

arXiv2022

A counterexample to the $ϕ$-dimension conjecture

In 2005, the second author and Todorov introduced an upper bound on the finitistic dimension of an Artin algebra, now known as the ϕ-dimension. The ϕ-dimension conjecture states that this upper bound is always finite, a fact that would imply the finitistic dimension conjecture. In this paper, we present a counterexample to the ϕ-dimension conjecture and explain where it comes from. We also discuss implications for further research and the finitistic dimension conjecture.

math.RT↗

arXiv2022

τ-Cluster Morphism Categories and Picture Groups

$τ$-cluster morphism categories, introduced by Buan and Marsh, are a generalization of cluster morphism categories (defined by Igusa and Todorov). We show the classifying space of such a category is a cube complex, generalizing results of Igusa and Todorov and Igusa. Furthermore, the fundamental group of this space is the picture group of the algebra, first defined by Igusa, Todorov, and Weyman. Finally, we show that for Nakayama algebras, this space is a $K(π,1)$. The key step is a combinatorial proof that, for Nakayama algebras, 2-simple minded collections are characterized by pairwise compatibility conditions, a fact not true in general.

math.RT↗

arXiv2022

Which cluster morphism categories are CAT(0)

The cluster morphism category of an hereditary algebra was introduced in [5] to show that the picture space of an hereditary algebra of finite representation type is a $K(π,1)$ for the associated picture group, thereby allowing for the computation of the homology of picture groups of finite type as carried out in [7] for the case of $A_n$. In this paper we show that the cluster morphism category is a $CAT(0)$-category for hereditary algebras of finite or tame type with only small tubes. As a consequence, we get that the classifying space of the cluster morphism category is a locally $CAT(0)$ space and, as a consequence of that, we get that this classifying space is a $K(π,1)$.

math.RT↗

arXiv2021

Second obstruction to pseudoisotopy in dimension 3

We use lens-shaped models and the second obstruction to pseudoisotopy to construct a nontrivial diffeomorphism of $M\times I$ where $M$ is the connected sum of $S^1\times S^2$ with a another nonsimply connected 3-manifold $M'$. Then we take two copies of this diffeomorphism and paste together their tops and bottoms to obtain a diffeomorphism of $M\times S^1$. Properties of the second obstruction and the first Postnikov invariant imply that this diffeomorphism of the closed 4-manifold $M\times S^1$ is not isotopic to the identity. Similar results were obtain by Singh [10].

math.GT↗

arXiv2021

Binary trees using the bookshelf and baseball constructions

This is a largely expository paper in which we discuss various sets having a Catalan number of objects and some well-known bijections between these sets presented in a new and hopefully interesting way. We call these concepts "bookshelf" and "baseball" constructions. No knowledge of these topics is assumed. These are the final student papers for a course at Brandeis University.

math.CO↗

arXiv2021

Continuous Quivers of Type A (III) Embeddings of Cluster Theories

We continue the work started in parts (I) and (II). In this part we classify which continuous type A quivers are derived equivalent and introduce the new continuous cluster category with E-clusters, which are a generalization of clusters. In the middle we provide a rigorous connection between the previous construction of the continuous cluster category and the new construction. We conclude with the introduction of a cluster theory, generalizing the notion of a cluster structure. Using this new notion, we demonstrate how one embeds known type A cluster theories into the new E-cluster theory in a way compatible with mutation. This is part (III) in a series of work that will conclude with a continuous generalization of mutation for cluster theories.

math.RT↗

arXiv2021

Second obstruction to pseudoisotopy I

A pseudoisotopy of $M$ is a diffeomorphism of $M\times I$ which is the identity on $M\times 0$. We give an explicit construction of pseudoisotopies of 4-manifolds which realize certain elements of the "second obstruction to pseudoisotopy". The next paper will give precise statements of our results for 5-manifolds which had been announced long ago.

math.GT↗

arXiv2021

Universal quivers

We show that for any positive integer $n$, there exists a quiver $Q$ with $O(n^2)$ vertices and $O(n^2)$ edges such that any quiver on $n$ vertices is a full subquiver of a quiver mutation equivalent to $Q$. We generalize this statement to skew-symmetrizable matrices and obtain other related results. In particular, we show that any quiver is a full subquiver of a quiver mutation equivalent to a quiver of a plabic graph.

math.CO↗

arXiv2021

Continuous quivers of type A (I) Foundations

We generalize type $A$ quivers to continuous type $A$ quivers and prove initial results about pointwise finite-dimensional (pwf) representations. We classify the indecomosable pwf representations and provide a decomposition theorem, recovering results of Botnan and Crawley-Boevey. We also classify the indecomposable pwf projective representations. Finally, we prove that many of the properties of finite-dimensional type $A_n$ representations are present in finitely generated pwf representations. This is the self-contained foundational part of a series of works to study a generalization of continuous clusters categories and their relationship to other type $A$ cluster structures.

math.RT↗

arXiv2020

A Legendrian Turaev torsion via generating families

We introduce a Legendrian invariant built out of the Turaev torsion of generating families. This invariant is defined for a certain class of Legendrian submanifolds of 1-jet spaces, which we call of Euler type. We use our invariant to study mesh Legendrians: a family of 2-dimensional Euler type Legendrian links whose linking pattern is determined by a bicolored trivalent ribbon graph. The Turaev torsion of mesh Legendrians is related to a certain monodromy of handle slides, which we compute in terms of the combinatorics of the graph. As an application, we exhibit pairs of Legendrian links in the 1-jet space of any orientable closed surface which are formally equivalent, cannot be distinguished by any natural Legendrian invariant, yet are not Legendrian isotopic. These examples appeared in a different guise in the work of the second author with J. Klein on pictures for $K_3$ and the higher Reidemeister torsion of circle bundles.

math.SG↗

arXiv2020

Continuously triangulating the continuous cluster category

In [4], the continuous cluster category was introduced. This is a topological category whose space of isomorphism classes of indecomposable objects forms a Moebius band. It was found in [4] that, in order to have a continuously triangulated structure on this category, one needs at least two copies of each indecomposable object forming a 2-fold covering space of the Moebius band. This paper classifies all continuous triangulations of finite coverings of the basic continuous cluster category. This includes the connected 2-fold covering of Igusa-Todorov [4], the disconnected 2-fold covering of Orlov [6] and a third unexpected continuously add-triangulated 2-fold covering of the Moebius strip category.

math.RT↗

arXiv2020

Resolution quiver and cyclic homology criteria for Nakayama algebras

If a Nakayama algebra is not cyclic, it has finite global dimension. For a cyclic Nakayama algebra, there are many characterizations of when it has finite global dimension. In [She17], Shen gave such a characterization using Ringel's resolution quiver. In [IZ92], the second author, with Zacharia, gave a cyclic homology characterization for when a monomial relation algebra has finite global dimension. We show directly that these criteria are equivalent for all Nakayama algebras. Our comparison result also reproves both characterizations. In a separate paper we discuss an interesting example that came up in our attempt to generalize this comparison result to arbitrary monomial relation algebras [HI19].

math.RT↗

arXiv2019

Continuous cluster categories II: continuous cluster-tilted categories

We show that the quotient of the continuous cluster category $\mathcal C_π$ modulo the additive subcategory generated by any cluster is an abelian category and we show that it is isomorphic to the category of infinite length modules over the endomorphism ring of the cluster. These theorems extend the theorems of Caldero-Chapoton-Schiffler and Buan-Marsh-Reiten for cluster categories to the continuous cluster category of type $A$. These results will be generalized in a series of forthcoming joint papers of the two authors with Job Rock.

math.RT↗