AlgebraiC Topology NETWORK

Description

Algebraic Topology Network, formerly known as Kansai Algebraic Seminar, organizes seminars and workshops and collects information about domestic seminars and workshops on algebraic topology for providing places where algebraic topologists based in Japan can communicate and exchange recent ideas and techniques around algebraic topology. 

Network Organizers

Sho Hasui (Osaka Metropolitan University)

Mitsunobu Tsutaya (Kyushu University)

Atsushi Yamaguchi (Osaka Metropolitan University)

Past Events

20 - 21 June

Homotopy Okinawa (PDF)

Venue

Okinawa Prefectural Gender Equality Center "Tiruru" (Homepage)

Celebration

We will celebrate Prof. Toshiyuki Akita's 60th birthday.

Confirmed Speakers

Toshiyuki Akita (Hokkaido Univrsity)

Yasuhiko Asao (Kyushu University)

Yoh Katoh (Tokyo University of Science)

Ye Liu (Xi’an Jiaotong-Liverpool University)

Takahiro Matsushita (Shinshu University)

Yuki Minowa (Kyushu University)

Takefumi Nosaka (Institute of Science Tokyo)

Takao Sato (Tokyo University of Science)

Ryokichi Tanaka (Kyoto University)

Schedule

20 June

9:10 - 10:00 Takahiro Matsushita

10:10 - 11:00 Takao Satoh

11:10 - 12:00 Yasuhiko Asao (special lecture)

Lunch break

14:00 - 14:50 Yasuhiko Asao (special lecture)

15:00 - 15:50 Ye Liu

16:00 - 16:50 Ryokichi Tanaka

17:00 - 17:50 Toshiyuki Akita

21 June

9:10 - 10:00 Takefumi Nosaka

10:10 - 10:50 Yuki Minowa

11:00 - 11:40 Yoh Katoh

Financial Support

JSPS KAKENHI 23K03113 (Takeshi Torii)

Organizers

Daisuke Kishimoto (Kyushu University)

Shuichi Tsukuda (University of the Ryukyus)

Abstract

Toshiyuki Akita

Wirtinger Presentations, Quandles, and Crossed Modules

Wirtinger presentations are well known as standard presentations of knot groups, but they are also closely related to quandles, free crossed modules, and the low-dimensional (co)homology of groups. In this talk, I will recall the notion of a Wirtinger presentation and then explain some connections among these topics, with an emphasis on my recent work.

Yasuhiko Asao

Magnitude theory as a homology theory of directed graphs 

In this two-part lecture series, we survey how directed graphs can be studied using methods from algebraic topology by regarding them as enriched categories, following the framework of Leinster’s magnitude theory, together with an introduction to the speaker’s contributions to the subject. In the first part, we treat posets, graphs, and directed graphs in a unified manner as enriched categories, and define magnitude as a generalization of the Euler characteristic of the nerve of a poset. We will see that various invariants fit naturally into this framework, including the Euler characteristic of simplicial complexes, the Poincaré polynomial of the intersection lattice of a hyperplane arrangement, and the growth series of finitely generated groups. 

In the second part, we define magnitude homology by generalizing the definition of homology as a categorification of Euler characteristic. We discuss its relationship with path homology and introduce work on the homotopy theory of graphs. We also explain a homological algebraic formulation of magnitude homology using Tor functors. 

If time permits, we will further discuss how these ideas extend to encompass persistent homology, and briefly mention persistent magnitude.

Yoh Katoh

Construction of Indecomposable Representations of the Automorphism Group of a Free Group 

The automorphism group of a free group has been studied for many years, partly because it contains topologically important groups such as mapping class groups and braid groups as subgroups. On the other hand, little is known about representations of the automorphism group of a free group that do not factor through the abelianization. In this talk, we consider the action of the automorphism group of a free group on the coordinate ring of the SL(k)-character variety of a free group, and construct finite-dimensional representations of the automorphism group using the maximal ideal associated with the trivial representation. We then show that, in the case k=2, the resulting representations are previously unknown indecomposable representations that do not factor through the abelianization. This talk includes a joint work with Takao Satoh.

Ye Liu

Magnitude homology of real hyperplane arrangements

Magnitude is a cardinality-like invariant of metric spaces or enriched categories measuring the effective size. Its categorification, the magnitude homology, is a more powerful invariant. For a real hyperplane arrangement, or more generally, an oriented matroid, the tope graph encapsulates considerable amount of information. Since tope graphs are equipped with the shortest path metric, we feed them to the magnitude and magnitude homology machinery to derive new invariants of real hyperplane arrangements. We prove some structural results of the magnitude of arrangements, including reciprocity, palindromic numerator and denominator. For magnitude homology of arrangements, we give combinatorial descriptions in small length and prove that tope graphs are diagonal if and only if the arrangement is Boolean. We present a face decomposition of magnitude homology, using which we obtain a combinatorial formula of diagonal magnitude Betti numbers. Many open problems are posted for future study. In particular, we conjecture that magnitude and magnitude homology of arrangements are determined by the intersection lattice.

Takahiro Matsushita

Closed neighborhood complex

The closed neighborhood complex N[G] of a simple graph G is the simplicial complex whose simplices are finite sets of vertices contained in a closed neighborhood of a vertex in G. We reveal that the closed neighborhood complex has close connections with other concepts, including the independence complex of the canonical double covering and the independence complex of the neighborhood hypergraph. Furthermore, we show that the fundamental group of the closed neighborhood complex is isomorphic to Grigor'yan--Lin--Muranov--Yau's fundamental group of a graph introduced in the study of path homology.

Yuki Minowa

Topological complexity sequences of groups

Topological complexity is a numerical homotopy invariant that measures the instability of motion planning in a space. In the study of this invariant, the case of K(π, 1)-spaces has been one of the core interests. I will present a new approach in this direction: a sequence of the topological complexities of the Milnor constructions of a group. I will also discuss results for finite groups, where it exhibits particularly intricate behavior. This talk is based on joint work with Daisuke Kishimoto.

Takefumi Nosaka

Q/Z-torsion in the third homology of diffeomorphism groups

The Smale conjecture provides a powerful principle connecting the homotopy type of the diffeomorphism group of a three-dimensional manifold with Lie-theoretic objects. Starting from this viewpoint, this talk explains how Chern–Simons-type invariants can be used to detect torsion in the third group homology of several diffeomorphism groups preserving geometric structures; we compare these diffeomorphism groups with compact Lie groups, and then restrict Chern–Simons-type 3-cocycles to cyclic subgroups. As a consequence, in each case we obtain an injection Q/Z into H_3^{gr}.

Takao Satoh

On the abelianizations of the derivation Lie algebras of free Lie algebras 

This is a joint work with Naoya Enomoto. In the 1980s, Shigeyuki Morita constructed the Morita traces to study the cokernels of the Johnson homomorphisms of the mapping class groups of surfaces. Now it is conjectured by Morita that the first Johnson homomorphism and the Morita traces completely describe the abelianization of the derivation Lie algebras of a free Lie algebra. This conjecture is shown in a stable range by Morita over the rational numbers, and by Morita-Sakasai-Suzuki over the integers. In this talk, we will talk about the images of the Morita traces restricted to the special derivation algebras of a free Lie algebra, and show that the abelianization of the special derivation algebra cannot be described by only the Morita traces. We will also talk on the abelianizations of the tangential derivations of  free Lie algebras.

Ryokichi Tanaka

Rough similarity rigidity via ergodic theory of topological flows

For every non-elementary hyperbolic group, we give a necessary and sufficient condition for two given word metrics to be roughly similar, i.e., they are within bounded distance after multiplying by a positive constant, in terms of mean distortion. The proof is based on the ergodic theory of topological flows associated with general hyperbolic groups. We will also mention explicit illustrative examples and applications, including the real analyticity of intersection numbers for families of dominated (Anosov) representations. This talk is based on joint work with Stephen Cantrell (St Andrews).

8 June 16:30--18:00

Shinshu Topology Seminar (Homepage)

池渕未来（京都大学）

Lawvere代数理論のホモロジーと等式論理

群の生成元と関係式が群を表示することと同様に，等式公理系はLawvere代数理論と呼ばれる小圏を表示することが知られている．さらに，群のホモロジーのランクが生成元や関係式の個数を下から抑えるのと同様に，与えられた等式公理系に対し，それと同値な等式公理系は少なくともいくつの公理からなるかは，Lawvere代数理論のホモロジーから計算できることが近年示された．この事実の特別な場合として，群の公理系と同値な任意の等式公理系(であって乗法の記号 _・_, 逆元の記号 _^{-1}，単位元の記号 e と変数から成るもの)は，少なくとも二つの等式を持つという，Tarskiの定理の別証明が得られる． この講演では上記の事実の紹介を行うとともに，Lawvere代数理論のホモロジーが一般的なQuillenのホモトピー代数の枠組みから定義できることや，Baues-Wirschingによる小圏のホモロジーとの関係などについて時間の限り述べる．

14 May 16:30--18:00

Shinshu Topology Seminar (Homepage)

折田龍馬（新潟大学）

Morse-Bott-Smale鎖複体

臨界点が全て非退化である関数をMorse関数という。特に、Morse関数の臨界点は孤立している。Morse関数は、その臨界点を生成元、臨界点を結ぶ勾配曲線の数え上げを微分として、鎖複体を定める。そのホモロジーを Morseホモロジーという。 一方、非退化性を緩め、臨界点集合が部分多様体を成しているような関数をMorse-Bott関数という。Morse-Bott関数は、一般に臨界点を無限個持つが、部分多様体のホモロジーを「載せる」ことにより、鎖複体を定めることができる。そのホモロジーをMorse-Bottホモロジーという。その定義の流儀は様々であるが、本講演では、Banyaga-Hurtubiseによる定義の「改善」について扱う。 本講演の内容は、矢代海音氏(新潟大学)との共同研究に基づく。

30 April 16:30--18:00

Shinshu Topology Seminar (Homepage)

浅尾泰彦（九州大学） 

Magnitude homology and Anick resolution

Hepworth-Willerton, Leinster-Shulmanによって与えられたマグニチュードホモロジーの定義は、マグニチュードの圏化となるようにデザインされた具体的な鎖複体のホモロジーであった。講演者の問題意識の一つとして、このホモロジーが既存の文脈の中でどう解釈されるべきか、というものがある。この講演ではAsao-IvanovによるマグニチュードホモロジーのTor関手としての記述に基づいて、「有限次元代数の表現論」的な解釈を与える試みについて紹介する。またAsao-Wakatsukiによる極小射影分解の構成を用いたマグニチュードホモロジーの計算を、Anick resolutionという表現論的な枠組みに沿って紹介する。この講演は一部アーロン・チャン氏 (名大)と若月駿氏（名大）とのディスカッションに基づく。

24 February 10:30-12:00

Kyushu University Topology Seminar

Le Minh Ha (Vietnam institute for Advanced Study in Mathematics)

Venue

Kyushu University, W1-D414

Modular Invariant Theory for Polynomial Rings mod Frobenius Powers

We will explain our solution to conjectures due to Lewis, Reiner, and Stanton concerning the Hilbert series of the invariant ring of a polynomial algebra modulo Frobenius powers, and discuss its consequences. This is joint work with Nguyen D. H. Hai, Nguyen V. Nghia, and Le X. Hoang.

21 January

Shinshu Topology Seminar (Homepage)

木村 満晃（大阪歯科大学）

相対単純群の坪井距離

川崎盛通氏、児玉大樹氏、松田能文氏、松下尚弘氏、折田龍馬氏との共同研究(arXiv:2412.00839)に基づき、相対単純群の概念およびその坪井距離について概説する。微分同相群などの様々な変換群が単純群であることは古典的に知られているが、その普遍被覆は必ずしも単純ではない。我々は単純群の自然な一般化として「相対単純群」という概念を導入し、様々な変換群の普遍被覆が相対単純群となることを観察した。また、微分同相群の研究の文脈において、坪井により単純群に対し定義された「坪井距離」は、相対単純群に対しても自然に定義される。本講演では、特にハミルトン微分同相群の場合について、相対単純性および坪井距離空間の擬等長類について論じる。