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

23 July 2025

Shinshu Topology Seminar (Homepage)

Venue

Science Building, Room A401, Shinshu University

Schedule

16:30 -- 18:00 Takumi Maekawa (The university of Tokyo)

A six-functor formalism and the Bauer-Furuta invarint

低次元トポロジーにおけるゲージ理論では、微分幾何・関数解析・代数トポロジーの手法を組み合わせることで、多様体の微分構造に関するさまざまな不変量が構成されてきた。なかでも、BauerとFurutaにより導入された不変量は、Seiberg-Witten理論に起源を持ちながら、同変球面の安定ホモトピー群に値をとるものとして知られる。近年、このBauer-Furuta不変量の族版を用いた四次元微分同相群のホモトピー型の研究が大きな注目を集めているが、その理論的背景について、代数トポロジーの観点からの総括は、十分に行われていない。本講演では、このような安定ホモトピー的な族ゲージ理論不変量が、安定無限大圏に値を取る層のsix-functor formalismに基づいて自然に構成されることを説明する。

7, 14 July 2025

Fukuoka University Topology Seminar (Homepage)

Venue

Science Building 9, Room 9433, Fukuoka University

Schedule

16:20 -- 17:50 Norio Iwase (Kyushu University)

単体圏の双対とMuro-Tonks homotopy associahedron

Stasheff の意味での空間の A∞構造は位相 monoid を up to homotopy で変形すると自然に現れるものであるが、見かけ上は strict 単位元の存在を必要としていた。岩瀬はこの strict 単位元の存在を A_2構造に関するhomootpy 単位元の存在まで緩めることに成功して Stasheff の初期の論文の予想を解決したのだが、その一方で、Muro-Tonks は Fukaya の A∞構造の位相空間版を記述するために、strict 単位元を homotopy 的に緩めた homotopy unital structure を与える homotopy unital associahedron を構成した。 さて、単体圏上の co-end として構成される位相 monoid の分類空間の Milnor 構成をモデルとして、 Stasheff は A∞空間の分類空間を構成しており、この構成は L-S カテゴリ数や位相的複雑さの決定に深く関与するものでもあった。ここでは homotopy unital A∞空間の分類空間の構成を目標としたい。その際に必要となるのは、Stasheff の associahedron を用いた単体圏の対応物の構成であり、この為に単体圏の双対圏を考えることが（自分には）必要であった。

28, 29 June 2025

Homotopy Okinawa 2025

Organizers

Daisuke Kishimoto, Shuichi Tsukuda

Venue

Okinawa Sailor's Hotel, Meeting Room 3

Support

JSPS KAKENHI JP22K03284 (Daisuke Kishimoto), JP23K03116 (Toshio Sumi), JP23K03113 (Takeshi Torii)

Schedule (PDF)

28 June

9:00 -- 10:00 Kensuke Arakawa (Kyoto University)

10:10 -- 11:10 Takahiro Matsushita (Shinshu University)

11:20 -- 11:50 Tomoki Tokuda (Kyushu University)

13:30 -- 14:30 Takuya Saito (Hokkaido University)

14:40 -- 15:40 So Yamagata (Fukuoka University)

15:40 --16:40 Takeshi Torii (Okayama University)

29 June

9:30 -- 10:30 Yuki Minowa (Kyoto University)

10:40 -- 11:40 Takahito Naito (Nippon Institute of Technolgy)

 Abstract

Kensuke Arakawa

Monoidal Relative Categories Model Monoidal ∞-Categories 

Homotopical study of mathematical objects often starts by identifying a subcategory of weak equivalences that behave like isomorphisms. In this spirit, relative categories offer a minimalistic framework for homotopy theory: They are categories equipped with a designated subcategory of weak equivalences. A remarkable theorem of Barwick and Kan shows that this simple structure in fact models ∞-categories. More precisely, any relative category (C,W) gives rise to an ∞-category C[W^{-1}] by localizing at the weak equivalences. This process defines a functor RelCat[DK^{-1}]—> Cat∞, where DK denotes the subcategory of relative functors that induce equivalences of localizations. Barwick and Kan showed that this functor is an equivalence. 

In practice, many relative categories come equipped with a monoidal structure whose tensor product preserves weak equivalences in each variable. In such cases, the localization inherits a monoidal structure. This raises a natural question: Do monoidal relative categories model monoidal ∞-categories? The author recently proved that the answer is yes. In this talk, we explain the ideas behind the proof, explore some applications, and suggest possible generalizations.

Takahiro Matsushita

Higher-dimensional generalization of Youngs' theorem and circular colorings

Quadrangulations of surfaces are regular CW structures on surfaces in which every 2-face is a square. These structures have been extensively studied in topological graph theory. A famous theorem by Youngs states that the chromatic number of the 1-skeleton of every quadrangulation of the real projective plane is 2 or 4. In this talk, by using methods of topological combinatorics, we obtain a higher-dimensional generalization of Youngs' theorem, which is a refinement involving circular chromatic numbers. This is joint work with Kengo Enami (Tsuda University).

Yuki Minowa

Rational sequential parametrized topological complexity

Sequential parametrized topological complexity is a numerical homotopy invariant of a fibration, which arose in the robot motion planning problem with external constraints. I will talk about sequential parametrized topological complexity in view of rational homotopy theory. I will give an explicit algebraic upper bound for sequential parametrized topological complexity when a fibration admits a certain decomposition, which is a generalization of the result of Hamoun, Rami and Vandembroucq on topological complexity. I will also mention a variant of the TC-generating function, which is introduced by Farber and Oprea.

Takahito Naito

Algebraic interleavings of spaces over the classifying space of the circle

The interleaving distance is a pseudometric on persistent objects values in a category, namely, objects in a functor category from R to a category. In this talk, we investigate the distance of persistent differential graded modules derived from spaces over the classifying space of the circle. We show that the distance coincides with variants of the interleaving distance in the homotopy category in the sense of Lanari and Scoccola and the homotopy interleaving distance in the sense of Blumberg and Lesnick. Moreover, we discuss the upper and lower bounds of this distance in relation to the cup-length of spaces over BS^1. This is joint work with K. Kuribayashi, S. Wakatsuki and T. Yamaguchi.

Takuya Saito

Towards the categorification of polynomial invariants in matroid theory I 

In this talk and the next we will talk about the categorification of invariants of matroids. Matroid theory was introduced independently by Whitney and Nakasawa. Matroids are structures obtained as an abstraction of dependencies, including linear dependencies and cycles in graphs. Graphs, hyperplane arrangements provide fundamental examples of matroids. In this talk, we will introduce the definition, examples, constructions, and invariants of matroids. This talk is based on the joint work with Yamagata.

Tomoki Tokuda

On the mod 2 cohomology of 2-configuration spaces

The n-configuration space C_n(X) is the space of all n-tuples of pairwise distinct points of X. Then the symmetric group acts on C_n(X) as permutations of coordinates. The orbit space B_n(X)=C_n(X)/Σ_n is called the unordered configuration space. The mod 2 cohomology of C_2(X) and B_2(X) are subjects of interest in the Borsuk--Ulam type theory. In this talk, we introduce some results and a method for computation of the mod 2 cohomology of C_2(X) and B_2(X) for the case of X is a torus or a closed surface.

Takeshi Torii

Map monoidales and duoidal ∞-categories

We have introduced a notion of duoidal ∞-category which is a generalization of the notion of duoidal category. A duoidal category has two monoidal products in which one is (op)lax monoidal with respect to the other. The goal of this talk is to give an example of duoidal ∞-category. 

A map monoidale is a pseudomonoid in a monoidal bicategory whose multiplication and unit are left adjoints. It is known that the endomorphism category of a map monoidale has the structure of a duoidal category. One monoidal product is given by composition and the other is given by convolution product. We introduce a convolution product for a map monoidale in a monoidal ∞-bicategory and show that the endomorphism ∞-category of a map monoidale has the structure of a duoidal ∞-category.

So Yamagata

Towards the categorification of polynomial invariants in matroid theory II 

Khovanov introduced a bigraded cohomology theory of links, whose graded Euler characteristic is the Jones polynomial. Analogously, several constructions of the Khovanov-type (co)homology theories have been provided beyond the knot theory, such as the chromatic cohomology of graphs and the characteristic homology of hyperplane arrangements. A matroid is a structure that reflects the notion of abstract dependency, including cycles in graphs and linear dependency of vectors. In particular, we can obtain matroids from both graphs and arrangements of hyperplanes. This talk is a continuation of "Towards the categorification of polynomial invariants in matroid theory I" by Saito.  In particular, we provide (co)homology groups associated with polynomial invariants in matroid theory as a generalization of the chromatic cohomology and characteristic homology of hyperplane arrangements. This talk is based on the joint work with Saito.

23 June 2025

Fukuoka University Topology Seminar (Homepage)

Venue

Science Building 9, Room 9433, Fukuoka University

Schedule

16:20 -- 17:50 Daisuke Kishimoto (Kyushu University)

Morse inequalities for noncompact manifolds

Morse inequalities relate the number of critical points of a Morse function and the Betti numbers of a closed manifold. Clearly, the compactness of a manifold is essential for Morse inequalities. In this talk, I will present Morse inequalities for noncompact manifolds having nice symmetry, where Morse functions are essentially irrelevant to the symmetry. Instead of the number of critical points and Betti numbers, we consider “configurations” of critical points and L^2-Betti numbers, and Morse inequalities are given in a certain preorder. As a corollary, we get the mean value version of Morse inequalities. 

This is joint work with Tsuyoshi Kato and Mitsunobu Tsutaya.

26 May 2025

Fukuoka University Topology Seminar (Homepage)

Venue

Science Building 9, Room 9433, Fukuoka University

Schedule

16:20 -- 17:50 Masaki Natori (The university of Tokyo)

A possible alternative proof of the real Bott periodicity theorem

Since Bott's original proof in 1959, several other proofs of complex Bott periodicity have been given. One such proof is due to Harris in 1980, which utilizes the spectral decomposition of unitary matrices and the group completion theorem. In this talk, we will discuss the possibility of applying this approach to (a part of) real Bott periodicity. This requires certain modifications, such as replacing the complex numbers with quaternions as the target space of the spectrum, and considering eigenspaces as real or quaternionic vector spaces.

12-16 May 2025

The East Asian Conference on Algebraic Topology (Homepage)

Venue

Hebei Normal University, Shijiazhuang, China

28 April 2025

Fukuoka University Topology Seminar (Homepage)

Venue

Science Building 9, Room 9433, Fukuoka University

Schedule

16:20 -- 17:50 Yuki Minowa (Kyoto University)

Rational sequential parametrized topological complexity

Sequential parametrized topological complexity is a numerical homotopy invariant of a fibration, which arose in the robot motion planning problem with external constraints. I will talk about sequential parametrized topological complexity in view of rational homotopy theory. I will give an explicit algebraic upper bound for sequential parametrized topological complexity when a fibration admits a certain decomposition, which is a generalization of the result of Hamoun, Rami and Vandembroucq on topological complexity. I will also mention a variant of the TC-generating function, which is introduced by Farber and Oprea.

30 April 2025

Shinshu University Topology Seminar (Homepage)

Venue

Science Building, Room A401, Shinshu University

Schedule

16:30 -- 17:30 Luigi Caputi (University of Bologna) 

Bridging between überhomology and double homology 

Überhomology is a recently defined triply-graded homology theory of simplicial complexes, which yields both topological and combinatorial information. When restricted to (simple) graphs, a certain specialization of überhomology gives a categorification of the connected domination polynomial at -1; which shows that it is related to combinatorial quantities. On the topological side, überhomology detects the fundamental class of homology manifolds, showing that this invariant is a mixture of both. From a more conceptual viewpoint, we will show that a specification of überhomology of simplicial complexes can be identified with the second page of the Mayer-Vietoris spectral sequence, with respect to the anti-star covers. As a corollary, this allows us to connect überhomology to the double homology of moment angle complexes as defined by Limonchenko-Panov-Song-Stanley. This is joint work with D. Celoria and C. Collari.

24 March 2025

Fukuoka University Topology Seminar (Homepage)

Venue

Science Building 9, Room 9433, Fukuoka University

Schedule

16:20 -- 17:50 Luigi Caputi (University of Bologna)

Bridging between überhomology and double homology 

Überhomology is a recently defined triply-graded homology theory of simplicial complexes, which yields both topological and combinatorial information. When restricted to (simple) graphs, a certain specialization of überhomology gives a categorification of the connected domination polynomial at -1; which shows that it is related to combinatorial quantities. On the topological side, überhomology detects the fundamental class of homology manifolds, showing that this invariant is a mixture of both. From a more conceptual viewpoint then, we show that überhomology of simplicial complexes can be identified with the second page of the Mayer-Vietoris spectral sequence, with respect to the anti-star covers. As a corollary, this allows us to connect überhomology to the double homology of a moment angle complex defined by Limonchenko-Panov-Song-Stanley.

5-7 March 2025

(Non)commutative Algebra and Topology (Homepage)

Venue

Shinshu University

29 January 2025

One Day Workshop with Rachael Boyd on Algebraic Topology

Organizer

Tsuyoshi Kato

Venue

Kyoto University, Science Bldg 3, Room 127

Schedule

9:30 -- 10:30 Tadayuki Watanabe (Kyoto University)

Brunnian links and Kontsevich graph complex

Ideas in low-dimensional topology are often powerful also for higher dimensional manifold bundles. For example, Goussarov-Habiro's theory of surgery on 1,3-valent graphs in 3-manifolds can be generalized for higher dimensional manifold bundles. Recently, we further generalized the 3-valent graph surgery of bundles to graphs with arbitrary higher valences to obtain a chain map GC^{(\leq l)}\to S_*(BDiff_\partial(D^{2k});Q) when 2k is sufficiently high. Here, GC^{(\leq l)} is the truncation of Kontsevich's graph complex GC to the submodule of graphs with excess l (may include graphs with (l+3)-valent vertices). We conjecture that it induces monomorphisms on homology up to excess l-1 and that the injectivity is detected by Kontsevich's configuration space integrals. Our generalized graph surgery is based on a family of Brunnian string links associated to a p-valent vertex that satisfies the L_\infty relation in the space of embeddings. This is a joint work with Boris Botvinnik.

10:45 --11:45 Daisuke Kishimoto (Kyushu University)

The fundamental group and the magnitude-path spectral sequence of a directed graph

The fundamental group of a directed graph admits a natural sequence of quotient groups called r-fundamental groups, and the r-fundamental groups can capture properties of a directed graph that the fundamental group cannot capture. The fundamental group of a directed graph is related to path homology through the Hurewicz theorem. The magnitude-path spectral sequence connects magnitude homology and path homology of a directed graph, and it may be thought of as a sequence of homology of a directed graph, including path homology. I will talk about relations of the r-fundamental groups and the magnitude-path spectral sequence through the Hurewicz theorem and the Seifert-van Kampen theorem.

 This is joint work with Yichen Tong.

13:30 -- 14:30 Kensuke Arakawa (Kyoto University)

Axiomatic approach to symmetric sequences and infinity operads

A symmetric sequence in a category C is a functor from the category of finite sets and bijections into C. When C is closed symmetric monoidal and cocomplete, the category of symmetric sequences inherits a monoidal structure with the composition product as its tensor product. Operads in C then correspond to monoids under this monoidal structure. 

One way to define operads in the infinity categorical setting is to generalize the composition product monoidal structure and define infinity operads as monoids in it. However, there are several, equally appealing ways to generalize the monoidal structure, and their comparison has proven difficult. It is therefore unclear whether these different generalizations lead to equivalent theories of infinity operads. 

In this talk, we show that the specific encoding of the composition product monoidal structure does not affect the resulting theory of infinity operads. More precisely, we show that any choice of the monoidal structure satisfying natural axioms leads to an equivalent theory of infinity operads.

14:45 -- 15:45 Toshiyuki Akita (Hokkaido University)

Wirtinger groups and quandles

Wirtinger presentations are generalizations of the presentations of knot and link groups, and Wirtinger groups are groups that have such presentations. Examples of Wirtinger groups include the fundamental groups of higher-dimensional knots and links, braid groups, Artin groups, Thompson groups, free crossed modules, and many others. Quandles are algebraic structures that axiomatically capture the properties of conjugation in a group. The associated group of a quandle is a Wirtinger group, and conversely, every Wirtinger group is the associated group of some quandle. In this talk, I will introduce some aspects of Wirtinger groups and quandles, and explain our results on their second homology.

16:00 -- 17:00 Rachael Boyd (University of Glasgow)

Diffeomorphisms of reducible 3-manifolds

I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space B Diff(M), for M a compact, connected, reducible 3-manifold. We prove that when M is orientable and has non-empty boundary, B Diff(M rel ∂M) has the homotopy type of a finite CW-complex. This was conjectured by Kontsevich and previously proved in the case where M is irreducible by Hatcher and McCullough.

Preprint