arXiv subjects

Kiyoshi Igusa

Publications and source records attributed to Kiyoshi Igusa.

At least 73 records · Page 4Linked to original sources

arXiv2013

Continuous cluster categories of type D

We construct continuous Frobenius categories of type $D$. The stable categories of these Frobenius categories are cluster categories which contain the standard cluster categories of type $D_n$. When $n=\infty$, maximal compatible sets of indecomposable objects are laminations of the punctured disk. Discrete laminations are clusters. This new construction is topological and it also gives an algebraic interpretation of the "tagged arcs" which occur in Schiffler's geometric description of clusters of type $D_n$.

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arXiv2013

Mixed cobinary trees

We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees. We show that the number of isomorphism classes of such trees is given by the Catalan number Cn where n is the number of internal nodes. We also consider the corresponding quiver Q_ε of type An-1. As a special case of more general known results about the relation between c-vectors, representations of quivers and their semi-invariants, we explain the bijection between mixed cobinary trees and the vertices of the generalized associahedron corresponding to the quiver Q_ε.

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arXiv2013

The first Hochschild cohomology group of a cluster-tilted algebra revisited

Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH1(B) with coefficients in the B-B-bimodule B. If C is a tilted algebra such that B is the relation extension of C by E= Ext2(DC,C), then we prove that HH1(B) is isomorphic, as a vector space, to the direct sum of HH1(C) with HH1(B,E). This yields homological interpretations for results of the first and the fourth author with M.J. Redondo.

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arXiv2013

Continuous Frobenius categories

We introduce continuous Frobenius categories. These are topological categories which are constructed using representations of the circle over a discrete valuation ring. We show that they are Krull-Schmidt with one indecomposable object for each pair of (not necessarily distinct) points on the circle. By putting restrictions on these points we obtain various Frobenius subcategories. The main purpose of constructing these Frobenius categories is to give a precise and elementary description of the triangulated structure of their stable categories. We show in arXiv:1209.1879 for which parameters these stable categories have cluster structure in the sense of [1] and we call these continuous cluster categories.

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arXiv2012

Continuous cluster categories I

In arXiv:1209.0038 we constructed topological triangulated categories C_c as stable categories of certain topological Frobenius categories F_c. In this paper we show that these categories have a cluster structure for certain values of c including c=pi. The continuous cluster categories are those C_c which have cluster structure. We study the basic structure of these cluster categories and we show that C_c is isomorphic to an orbit category D_r/F_s of the continuous derived category D_r if c=r pi/s. In C_pi, a cluster is equivalent to a discrete lamination of the hyperbolic plane. We give the representation theoretic interpretation of these clusters and laminations.

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arXiv2012

Exotic smooth structures on topological fibre bundles II

We use a variation of a classical construction of A. Hatcher to construct virtually all stable exotic smooth structures on compact smooth manifold bundles whose fibers have sufficiently large odd dimension (at least twice the base dimension plus 3). Using a variation of the Dwyer-Weiss-Williams smoothing theory which we explain in a separate joint paper with Bruce Williams [11], we associate a homology class in the total space of the bundle to each exotic smooth structure and we show that the image of this class in the homology of the base is the Poincaré dual of the relative higher Igusa-Klein (IK) torsion invariant. This answers the question, in the relative case, of which cohomology classes can occur as relative higher torsion classes.

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arXiv2012

Exotic smooth structures on topological fibre bundles I

When two smooth manifold bundles over the same base are fiberwise tangentially homeomorphic, the difference is measured by a homology class in the total space of the bundle. We call this the relative smooth structure class. Rationally and stably, this is a complete invariant. We give a more or less complete and self-contained exposition of this theory which is a reformulation of some of the results of [7]. An important application is the computation of the Igusa-Klein higher Reidemeister torsion invariants of these exotic smooth structures. Namely, the higher torsion invariant is equal to the Poincaré dual of the image of the smooth structure class in the homology of the base. This is proved in the companion paper [11] written by the first two authors.

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arXiv2011

Twisting cochains and higher torsion

This paper gives a short summary of the central role played by Ed Brown's "twisting cochains" in higher Franz-Reidemeister (FR) torsion and higher analytic torsion. Briefly, any fiber bundle gives a twisting cochain which is unique up to fiberwise homotopy equivalence. However, when they are based, the difference between two of them is a higher algebraic K-theory class measured by higher FR torsion. Flat superconnections are also equivalent to twisting cochains.

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arXiv2011

A proof of the strong no loop conjecture

The strong no loop conjecture states that a simple module of finite projective dimension over an artin algebra has no non-zero self-extension. The main result of this paper establishes this well known conjecture for finite dimensional algebras over an algebraically closed field.

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arXiv2009

Exceptional sequences and clusters

We show that exceptional sequences for hereditary algebras are characterized by the fact that the product of the corresponding reflections is the inverse Coxeter element in the Weyl group. We use this result to give a new combinatorial characterization of clusters tilting sets in the cluster category in the case where the hereditary algebra is of finite type.

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arXiv2009

Iterated integrals of superconnections

Starting with a Z-graded superconnection on a graded vector bundle over a smooth manifold M, we show how Chen's iterated integration of such a superconnection over smooth simplices in M gives an A-infinity functor if and only if the superconnection is flat. If the graded bundle is trivial, this gives a twisting cochain. Very similar results were obtained by K.T. Chen using similar methods. This paper is intended to explain this from scratch beginning with the definition and basic properties of a connection and ending with an exposition of Chen's "formal connections" and a brief discussion of how this is related to higher Reidemeister torsion.

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arXiv2009

Families of regular matroids

This is an introductory paper about the category of regular oriented matroids (ROMs). We compare the homotopy types of the categories of regular and binary matroids. For example, in the unoriented case, they have the same fundamental group but we show that the higher homotopy groups are different for rank three regular and binary matroids. We also speculate on the possible impact of a recent theorem of Galatius [Gal] computing the stable cohomology of the category of graphs and on possible applications to higher Reidemeister torsion.

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arXiv2008

Cluster Complexes via Semi-Invariants

We define and study virtual representation spaces having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the First Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition Theorem. In the special case of Dynkin quivers with n vertices this gives the fundamental interrelationship between supports of the semi-invariants and the Tilting Triangulation of the (n-1)-sphere.

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arXiv2005

Axioms for higher torsion invariants of smooth bundles

We explain the relationship between various characteristic classes for smooth manifold bundles known as ``higher torsion'' classes. We isolate two fundamental properties that these cohomology classes may or may not have: additivity and transfer. We show that higher Franz-Reidemeister torsion and higher Miller-Morita-Mumford classes satisfy these axioms. Conversely, any characteristic class of smooth bundles satisfying the two axioms must be a linear combination of these two examples. We also show how higher torsion invariants can be computed using only the axioms. Finally, we explain the conjectured formula of S. Goette relating higher analytic torsion classes and higher Franz-Reidemeister torsion.

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arXiv2005

Outer authomorphisms and the Jacobian

A graphs of rank n (homotopy equivalent to a wedge of n circles) without ``separating edges'' has a canonical n-dimensional compact C^1 manifold thickening. This implies that the canonical homomorphism f:Out(F_n)-> GL(n,Z) is trivial in rational cohomology in the stable range answering a question raised by Hatcher and Vogtmann [6]. Another consequence of the construction is the existence of higher Reidemeister torsion invariants for IOut(F_n)=ker f. These facts were first proved by the first author in [8] using different methods.

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arXiv2004

Combinatorial Miller-Morita-Mumford classes and Witten cycles

We obtain a combinatorial formula for the Miller-Morita-Mumford classes for the mapping class group of punctured surfaces and prove Witten's conjecture that they are proportional to the dual to the Witten cycles. The proportionality constant is shown to be exactly as conjectured by Arbarello and Cornalba [J. Alg. Geom. 5 (1996) 705-749]. We also verify their conjectured formula for the leading coefficient of the polynomial expressing the Kontsevich cycles in terms of the Miller-Morita-Mumford classes.

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arXiv2004

Increasing trees and Kontsevich cycles

It is known that the combinatorial classes in the cohomology of the mapping class group of punctures surfaces defined by Witten and Kontsevich are polynomials in the adjusted Miller-Morita-Mumford classes. The leading coefficient was computed in [Kiyoshi Igusa: Algebr. Geom. Topol. 4 (2004) 473-520]. The next coefficient was computed in [Kiyoshi Igusa: math.AT/0303157, to appear in Topology]. The present paper gives a recursive formula for all of the coefficients. The main combinatorial tool is a generating function for a new statistic on the set of increasing trees on 2n+1 vertices. As we already explained in the last paper cited this verifies all of the formulas conjectured by Arbarello and Cornalba [J. Alg. Geom. 5 (1996) 705--749]. Mondello [math.AT/0303207, to appear in IMRN] has obtained similar results using different methods.

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arXiv2003

Graph cohomology and Kontsevich cycles

The dual Kontsevich cycles in the double dual of associative graph homology correspond to polynomials in the Miller-Morita-Mumford classes in the integral cohomology of mapping class groups. I explain how the coefficients of these polynomials can be computed using Stasheff polyhedra and results from my previous paper GT/0207042.

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