igusa at brandeis

• Gereralized Grassmann invariant-redrawn v2

  • arXiv 2502,19147

  • This is my old unpublished paper called "The generalized Grassmann invariant". It shows how "pictures" also known as "Peiffer diagrams" represent elements of H3G for any group G and shows that K3(ℤ[G]) is isomorphic to a group of deformation classes of pictures for the Steinberg group of ℤ[G]. A picture representing an element of order 16 in K3(ℤ)≅ℤ48 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 An. 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.

• Ghost modules II

  • We use ``extension ghosts'' to obtain new formulas for invariants on K3 of integer groups rings.

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

  • arXiv:2303.14892

•  Second obstruction to pseudoisotopy in dimension 3  

  • arXiv 2112.08293.     

  • We use lens-shaped models and the second obstruction to pseudoisotopy to construct a nontrivial diffeomorphism of MxI where M is the connected sum of S1 x S2 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 MxS1. Properties of the second obstruction and the first Postnikov invariant imply that this diffeomorphism of the closed 4-manifold MxS1 is not isotopic to the identity. Similar results were obtain by Singh [10].

• Second obstruction to pseudoisotopy I  

  • arXiv 2110.09659.     

  • Given a smooth 4-manifold M with an embedded 2-sphere having odd self-intersection number we show that every element of the second obstruction group Wh1+(π1M;Z2) is realized in π0C(M), the concordance space of M. This is an updated version of old notes I have. The next two papers will deal with 3- and 5-manifolds.

• A Legendrian Turaev torsion via generating families[formerly "Turaev torsion of ribbon Legendrians"] (with Daniel Álvarez-Gavela)  

  •  Journal de l'École polytechnique - Mathematics, Volume 8 (2021), pp. 57-119.     

  • 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 K3 and the higher Reidemeister torsion of circle bundles.

•  An equivariant version of Hatcher's G/O construction. (with Tom Goodwillie and Chris Ohrt)  

  • J Topology (2015) 8 (3): 675-690.      

  • This paper gives an explicit construction of the generators of the rational homotopy groups of the space of stable h-cobordisms of the classifying space of a cyclic group of order n and calculate the higher torsion of these bundle verifying that the cohomology of these spaces is generated by equivariant higher torsion invariants. This plays an important role in Chris Ohrt's axiomatization of the higher equivariant torsion invariants.

•  Exotic smooth structures on topological fibre bundles I. (with Sebastian Goette and Bruce Williams)   

  • ArXiv:1203.2203   Trans. Amer. Math. Soc. 366 (2014) 749-790.   

  • This paper used to be the appendix of the Exotic smooth structures paper. It is now a separate paper at the suggestion of the referee. Given a smooth bundle, an exotic smooth structure is a smooth structure different from the one already given. Such structures are classified, rationally and stably, by a homology class in the total space of the bundle. We call it the relative smooth structure class.

• Exotic smooth structures on topological fibre bundles II. (with Sebastian Goette)   

  • ArXiv:1011.4653   Trans. Amer. Math. Soc. 366 (2014) 791-832.    

  • We use a variation of Hatcher's construction to construct virtually all stable exotic smooth structures on compact smooth manifold bundles whose fibers have sufficiently large odd dimension and and show that the relative higher Reidemeister torsion is the Poincare dual of the image of the relative smooth structure class (constructed in Part B) in the homology of the base.

•  Iterated integrals of superconnections.   

  • ArXiv:0912.0249   December 1, 2009   

•  Twisting cochains and higher torsion.  

  •  arXiv:0212383v3  Journal of Homotopy and Related Structures. 6 (2011), no.2, 213-238.

• Families of regular matroids.  

  •  ArXiv:0911.2014   November 11, 2009   

• K. Igusa, K. Orr, G. Todorov, J. Weyman. "Modulated quivers, semi-invariant pictures and picture groups"

  • The book is finally finished!

  • submitted to Cambridge University Press

  • Abstract: Pictures, also known as "spherical diagrams”, arise in topology, in particular algebraic K-theory and link invariants. But the authors found that they are also related to cluster theory and semi-invariants for representations of modulated quiver. In this book we develop the theory from scratch. We start with modulated quivers and semi-invariants which are defined when dimension vectors of modules satisfy certain stability conditions. These "domains of semi-invariants” forms the "semi-invariant picture”. In Chapter 3, these give rise to the "picture space” for the quiver and we prove it is a K(G,1) for the "picture group” using HNN extensions and "graphs of groups”. This allows us to compute the cohomology of the picture group of type An. Exceptional sequences and cluster theory, reviewed in the introduction are combined to give the "cluster morphism category”. The original paper introducing this concept is included as Chapter 4 of this book. This introduces "signed exceptional sequences” and shows they are in bijection with ordered clusters. Finally, in Chapter 5 we present periodic trees and show how they give rise to "propictures” in the case of affine quivers of type An.

• K. Igusa and R. Maresca. ``The Hom-Ext quiver and applications to exceptional collections''

  • arXiv:2509.16388

  • Abstract: We study what we call the Hom-Ext quiver and characterize it as a type of `superquiver'. In affine type 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 affine type A~, and show that the Hom-Ext quiver classifies exceptional sets up to derived autoequivalence. 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.

• Igusa, Kiyoshi, and Emre Sen. ``Exceptional sequences of type Bn/Cn and those in the abelian tube'' 

  • arXiv:2310.01700.v3

  • Abstract: 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 Bn/Cn, 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 Bn/Cn with maximal rigid objects in the cluster tube of rank n+1. The bijection passes through the set of ``augmented'' rooted labeled trees.

• K. Igusa and G. Todorov: ``Short history of signed exceptional sequences''

  • arXiv:2509.10910

  • Abstract: Signed exceptional sequences arose from the study of ``pictures'', ``picture groups'' and ``picture spaces''. The broad outline is as follows. In the 1970’s, for my PhD thesis work, I developed the concept of a ``picture'' and used it to define algebraic K-theory invariants for diffeomorphisms of smooth manifolds. Later, Kent Orr and I used pictures to settle a conjecture about link invariants. Around the turn of the century, Fomin and Zelevinsky developed the theory of cluster algebras and Todorov, together with Buan, Marsh, Reineke and Reiten related this to tilting theory. I realized that this was related to my ``pictures''. Jerzy Weyman realized that cluster-tilting related to his work with Harm Derksen on ``semi-invariants". This paper explains how this developed into the concept of ``semi-invariant pictures'', ``picture spaces'' and Signed Exceptional Sequences.

• More ghost modules I

  • arXiv:2506.129042506.12904:2506.129042506.12904

  • Abstract. Ghost modules were introduced in arXiv 2502,19147 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 basis 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 the higher Reidemeister torsion carried out in [IK].

• Ghost modules III

  • not ready.