arXiv subjects

Kiyoshi Igusa

Publications and source records attributed to Kiyoshi Igusa.

At least 19 recordsLinked to original sources

arXiv2026

Legendrian and Lagrangian higher torsion

Let $M$ be a closed manifold. We introduce a family of Legendrian isotopy invariants for Legendrians in $J^1M$, which we collectively call Legendrian higher torsion. Given a choice of a class $\mathcal{F}$ of fibre bundles over $M$, equipped with suitable unitary local systems, the Legendrian higher torsion of a Legendrian $Λ\subset J^1M$ is the subset of $H^*(M;\mathbf{R})$ consisting of higher Reidemeister torsion cohomology classes of fibre bundles $W$ over $M$ in the class $\mathcal{F}$ such that $Λ$ admits a generating function on a stabilization of $W$. For the class of tube bundles in the sense of Waldhausen we call the invariant tube torsion. We show that the tube torsion of a nearby Lagrangian $L \subset T^*M$ is well-defined when the stable Gauss map $L \to U/O$ is trivial (for example when $L$ is a nearby Lagrangian homotopy sphere) and it consists of a union of cosets of a normalized version of the Pontryagin character. We also identify a distinguished coset, invariant under Hamiltonian isotopy of $L$, which we call nearby Lagrangian torsion. We do not know whether nearby Lagrangians must have trivial tube torsion, as would follow from the nearby Lagrangian conjecture. However, we show that there exist Legendrians $Λ\subset J^1M$ with nontrivial tube torsion whose projection $Λ\to M$ is homotopic to a diffeomorphism.

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arXiv2026

Balanced notation for $τ$-rigid pairs

We introduce a new notation for $τ$-rigid pairs called ``balanced pairs''. The notation $\frac AB$ allows for simplification of some formulas, for example the Jasso category and its dual. Using this balanced notation we show that the $g$-vector $g(\frac AB)$ has nice geometric and algebraic implications: It defines ``lower'' and ``upper'' chambers and we prove that this corresponds to generalized Bongartz and co-Bongartz completions of balanced pairs. We also show that the balanced notation agrees the wall labels for the semi-invariant picture in the hereditary case and, in the general case, we describe the relation between balanced notation and the wall labels.

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arXiv2026

Pseudo-torsion classes

For a finite dimensional algebra $Λ$, we consider a torsion class $G$ in $mod$-$Λ$, which is not necessarily finitely generated. We construct a wall-and-chamber structure for $G$ where the chambers are the connected components of the complement of the union of walls. We also consider ``infinitesimal chambers". To each chamber we associate a ``pseudo-torsion class'' and a ``pseudo-torsionfree class'' and show that they are all distinct. We consider ``green paths'' in the stability space and associate to them Harder-Narasimhan stratifications of $G$. This paper is part of a series of papers whose goal is to study the ``ghosts'' which are remnants of the indecomposable $Λ$-modules which do not lie in $G$. In the special case when our torsion class is all of $mod$-$Λ$, we are in the classical well-known setting. All of our results apply to this classical setting. The ``pseudo-torsion classes'' are torsion classes. We point out that we do not take the closure of the set of walls. So, we get more chambers and our results are new even in this classical setting.

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arXiv2026

The Hom-Ext quiver and applications to exceptional collections

We study what we call the Hom-Ext quiver and characterize it as a type of `superquiver'. In type $\tilde{\mathbb{A}}$, the Hom-Ext quiver of an exceptional set is the tiling algebra of the corresponding geometric model. And, in that case, Hom-Ext quivers classify exceptional sets up to Dehn twist of the corresponding geometric model. We show that these Dehn twists are realized by twist functors and give autoequivalences of the derived category. We provide a generating set for the group of autoequivalences of the derived category in type $\tilde{\mathbb{A}}$, and show that the Hom-Ext quiver classifies exceptional sets up to the action of the subgroup of the automorphism group of the derived category generated by twist functors associated to exceptional cycles. We introduce superquivers, which are a generalization of Hom-Ext quivers. Exceptional sets over finite acyclic quivers are realized as representations of superquivers. Throughout, we list several questions and conjectures that make for, what we believe, exciting new research.

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arXiv2026

Short history of signed exceptional sequences

Whereas exceptional sequences have a long history with many well-known connections to combinatorics, signed exceptional sequences are relatively recent. The authors introduced this concept in 2017 [19], although it was retroactively realized that the category of noncrossing partitions [24] is a special case of this construction. Buan and Marsh [4] have introduced the concept of $τ$-exceptional sequences to generalize the definitions and theorems to all finite dimensional algebras. This short paper is the story of the original concept of signed exceptional sequences for hereditary algebras and how it developed out of the two authors' study of algebraic K-theory, link invariants, and cluster combinatorics.

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arXiv2025

Exceptional sequences of type $B_n/C_n$ and those in the abelian tube

We examine clusters in the cluster tube of rank $n+1$ using exceptional sequences in the abelian tube of rank $n+1$. Although the abelian tube has more exceptional sequences than the module categories of type $B_{n}/C_{n}$, we obtain a bijection between the set of signed exceptional sequences of any length in these categories. This bijection gives a reinterpretation of the formula of Buan-Marsh-Vatne comparing clusters of type $B_n/C_n$ with maximal rigid objects in the cluster tube of rank $n+1$. The bijection passes through the set of "augmented" rooted labeled trees.

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arXiv2025

More ghost modules I

Ghost modules were introduced in [I3] without definitions or proofs. We also introduced stability diagrams or "relative pictures" for torsion classes and torsion-free classes for representations of Dynkin quivers. Modules which were not in the chosen class reappeared as "ghosts", in fact one missing module produced two ghosts. In this short paper, we give a precise definition of ghost modules. We give several examples and prove basic properties of ghosts and pictures for torsion and torsion-free classes. We also introduce a third kind of ghost which we call "extension ghosts". In the next paper we will explain how these new ghosts can be used to visualize the computation of other invariants of $K_3$ of group rings.

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arXiv2025

Picture groups and maximal green sequences

We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of "vertically and horizontally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type). Furthermore, we show that every picture for such a set of positive roots is a linear combination of "atoms" and we give a precise description of atoms as special semi-invariant pictures.

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arXiv2025

On the functoriality of the space of equivariant smooth $h$-cobordisms

We construct an $(\infty,1)$-functor that takes each smooth $G$-manifold with corners $M$ to the space of equivariant smooth $h$-cobordisms ${\mathcal H}_{\mathrm{Diff}}(M)$. We also give a stable analogue ${\mathcal H}^{\mathcal U}_{\mathrm{Diff}}(M)$ where the manifolds are stabilized with respect to representation discs. The functor structure is subtle to construct, and relies on several new ideas. In the non-equivariant case $G=e$, our $(\infty,1)$-functor agrees with previous constructions of the smooth $h$-cobordism space as a functor to the homotopy category.

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arXiv2025

Derived delooping levels and finitistic dimension

In this paper, we develop new ideas regarding the finitistic dimension conjecture, or the findim conjecture for short. Specifically, we improve upon the delooping level by introducing three new invariants called the effective delooping level $\mathrm{edell}$, the sub-derived delooping level $\mathrm{subddell}$, and the derived delooping level $\mathrm{ddell}$. They are all better upper bounds for the opposite Findim. Precisely, we prove \[ \mathrm{Findim}\,Λ^{\mathrm{op}} = \mathrm{edell}\,Λ\leq \mathrm{ddell}\,Λ\text{ (or $\mathrm{subddell}\,Λ$)} \leq \mathrm{dell}\,Λ\] and provide examples where the last inequality is strict (including the recent example from [16] where $\mathrm{dell}\,Λ=\infty$, but $\mathrm{ddell}\, Λ= 1 =\mathrm{Findim}\, Λ^{\mathrm{op}}$). We further enhance the connection between the findim conjecture and tilting theory by showing finitely generated modules with finite derived delooping level form a torsion-free class $\mathcal{F}$. Therefore, studying the corresponding torsion pair $(\mathcal{T}, \mathcal{F})$ will shed more light on the little finitistic dimension. Lastly, we relate the delooping level to the $ϕ$-dimension $ϕ\dim$, a popular upper bound for findim, and give another sufficient condition for the findim conjecture.

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arXiv2025

Generalized Grassmann invariant-redrawn

This is my old unpublished paper called "The generalized Grassmann invariant". It shows how "pictures" also known as "Peiffer diagrams" represent elements of $H_3G$ for any group $G$ and shows that $K_3(\mathbb Z [G])$ is isomorphic to a group of deformation classes of pictures for the Steinberg group of $\mathbb Z[G]$. A picture representing an element of order $16$ in $K_3(\mathbb Z)\cong \mathbb Z_{48}$ is also constructed. In this updated version of the paper, we modify only the pictures and leave the text more or less unchanged. We also added an Appendix to explain the new pictures using representations of quivers and root systems of type $A_n$. Often, some roots are missing in the Morse pictures. We give two ideas to replace these roots. One uses "ghost handle slides" to obtain a standard picture. The second idea uses the (real) Cartan subalgebra $H$ to obtain a "relative" picture for a torsion class and adds "ghost modules" which are directly related to the generalized Grassmann invariant. Additions and changes are in blue except the pictures are black with colored ghosts.

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arXiv2025

A category of noncrossing partitions

In [17], we introduced ``picture groups'' and computed the cohomology of the picture group of type $A_n$. This is the same group what was introduced by Loday [20] where he called it the ``Stasheff group''. In this paper, we give an elementary combinatorial interpretation of the {\color{blue}``cluster morphism category'' constructed in [13] in the special case of the linearly oriented quiver of type $A_n$.} We prove that the classifying space of this category is locally $CAT(0)$ and thus a $K(π,1)$. We prove a more general statement that classifying spaces of certain ``cubical categories'' are locally $CAT(0)$. The objects of our category are the classical noncrossing partitions introduced by Kreweras [19]. The morphisms are binary forests. This paper is independent of [13] and [17] except in the last section where we use [13] to compare our category with the category with the same name given by Hubery and Krause [9].

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arXiv2024

Exceptional sequences and rooted labeled forests

We give a representation-theoretic bijection between rooted labeled forests with $n$ vertices and complete exceptional sequences for the quiver of type $A_n$ with straight orientation. The ascending and descending vertices in the forest correspond to relatively injective and relatively projective objects in the exceptional sequence. We conclude that every object in an exceptional sequence for linearly oriented $A_n$ is either relatively projective or relatively injective or both. We construct a natural action of the extended braid group on rooted labeled forests and show that it agrees with the known action of the braid group on complete exceptional sequences. We also describe the action of $Δ$, the Garside element of the braid group, on rooted labeled forests using representation theory and show how this relates to cluster theory.

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arXiv2024

Bijection between positive clusters and projectively signed exceptional sequences

In 2017, Igusa and Todorov gave a bijection between signed exceptional sequences and ordered partial clusters. In this paper, we show that every term in an exceptional sequence is either relatively projective or relatively injective or both and we refine this bijection to one between projectively signed exceptional sequences and ordered partial positive clusters. We also give a characterization of relatively projective/injective objects in terms of supports of the objects in the exceptional sequence.

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arXiv2024

Infinitesimal semi-invariant pictures and co-amalgamation

The purpose of this paper is to study the local structure of the semi-invariant picture of a tame hereditary algebra near the null root. Using a construction that we call co-amalgamation, we show that this local structure is completely described by the semi-invariant pictures of a collection of self-injective Nakayama algebras. We then describe the cones of this local structure using cluster-like structures that we call support regular clusters. Finally, we show that the local structure is (piecewise linearly) invariant under cluster tilting.

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arXiv2024

Generalized Goulden-Yong duals and signed minimal factorizations

We show the equivalence between one-way reflections and relative projective representations. We construct generalized Goulden-Yong duals using reverse Garside element actions and folded chord diagrams. We give two applications of the generalized Goulden-Yong duals: constructing generalized Prüfer codes and counting signed factorizations using the matrix-tree theorem.

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arXiv2024

Enumerating m-clusters using exceptional sequences

We give a bijection between ordered $m$-clusters and (complete) $m$-exceptional sequences, a concept that we introduce for this purpose. This holds for all hereditary artin algebras. This extends the bijection in the $m = 1$ case shown in arXiv:1706.02041.

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arXiv2023

Probability distribution for exceptional sequences of type $A_n$

We determine the probability distribution for relative projective objects in an exceptional sequence of type $A_n$ of any length. We show that these events (the $j$-th object in an exceptional sequence of length $k\le n$ being relatively projective) are independent of each other and from the length of the sequence. This gives a probabilistic interpretation of the product formula for the number of exceptional sequences of length $k$ and clusters or partial clusters of size $k$ since the latter numbers are proportional to the number of signed exceptional sequences of length $k$.

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