20 -- 21 December 2025
Fukuoka Homotopy Theory Seminar (Homepage)
Venue
Science Building 18, Room 1824, Fukuoka University
17 November 2025
Fukuoka University Topology Seminar (Homepage)
Venue
Science Building 9, Room 9433, Fukuoka University
Schedule
13:00 -- 14:00 Ryo Matsuda (Ritsumeikan University)
Renormalized volume and Teichmüller theory
We study the renormalized volume of quasi-Fuchsian manifolds of the 4-punctured sphere. Numerical comparisons are made with entropy of pseudo-Anosov, volume of mapping torus, and Teichmuller distance. In this talk, I will begin with the definition of the Teichmüller space, and focus on the relationship between two- and three-dimensional hyperbolic geometry. We also discuss a new distance defined by Masai and related perspectives. This talk is based on joint work with Hidetoshi Masai (Musashino Art University).
14:30 -- 15:30 Daisuke Kishimoto (Kyushu University)
Uniform Lefschetz fixed-point theorem
I will talk about joint work with Tsuyoshi Kato and Mitsunobu Tsutaya on the Lefschetz fixed-point theory for noncompact manifolds. We define the Lefschetz class of a uniformly continuous self-map of a noncompact manifold of bounded geometry, which stays within a bounded distance from the identity map, as an element of Block and Weinberger’s uniformly finite homology. We then prove that the Lefschetz class is zero if and only if the map is uniformly homotopic to a strongly fixed-point free map. To achieve this, we introduce a new cohomology for metric spaces, called uniform bounded cohomology, and develop an obstruction theory based on it.
16:20 -- 17:20 Yanga Bavuma (University of Cape Town)
Embeddings of locally compact abelian p-groups in Hawaiian groups
Locally compact abelian groups are rich in commuting closed subgroups. As a step in an attempt to find a geometric interpretation of the classification of locally compact groups we show that locally compact abelian p-groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. The construction we used was by Przezdziecki as a simplification of Virk's original answer to the question: for which groups is it possible to construct a path connected compact Hausdorff space whose fundamental group gives the group itself? It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring.
20 October 2025
Fukuoka University Topology Seminar (Homepage)
Venue
Science Building 9, Room 9433, Fukuoka University
Schedule
16:20 -- 17:20 Takuya Saito（Hokkaido University）
部分空間配置，重み付き半順序集合，畳み込み公式
超平面配置やマトロイドの特性多項式は，組合せ的・トポロジー的な様々な情報を持った重要な不変量である．特性多項式に対して制限と縮約を用いた畳み込み公式（Kung 2004）が知られている．本講演では超平面配置を一般化した部分空間配置に対して，その特性多項式の畳み込み公式を与える．また，色付き超グラフの彩色多項式への応用を紹介する．さらに（ポリ）マトロイドを含む重み付き半順序集合への一般化を考える．
24 -- 26 October 2025
Homotopy Theory Symposium 2025 (Homepage)
Venue
Science Building A, Room A-401, Shinshu University (24 Oct)
Science Lecture Room, Shinshu University (25, 26 Oct)
27 -- 28 September 2025
Okayama Autumn Conference 2025
~(Un)stable, Chromatic homotopy theory and arithmetic~
(Homepage )
Venue
Science Building 2, Room D201, Okayama University
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.
9 November 2024
Special Workshop
on the occasion of Atsushi Yamaguchi's 65th birthday
Organizers
Sho Hasui, Takefumi Nosaka
Venue
Institute of Science Tokyo, Ookayama Campus, Main Building, Room H213 (The northeast entrance of the building is open, Room H213 is on your right if you go up the stairs at the northeast entrance.)
Schedule
10:00 -- 10:50 Yuki Minowa (Kyoto University)
Rational parametrized topological complexity
The parametrized topological complexity is a variant of the topological complexity that is defined on a fibration p : E --> B. It was first introduced by Cohen, Farber and Weinberger to formulate the robot motion planning problem which involves external constraints. In this talk, we will consider the rationalization of this invariant, which is characterized in terms of a relative Sullivan model of p : E --> B. We will give some upper and lower bounds for the rational parametrized topological complexity. We will also mention a variant of the TC-generating function conjecture raised by Farber and Oprea.
11:00 -- 11:50 Katsuhiko Kuribayashi (Shishu University)
Interleavings of spaces over the classifying space of the circle
We bring spaces over the classifying space BS^1 of the circle group to persistence theory via the cohomology with coefficients in a field. Then, the cohomology interleaving distance between spaces over BS^1 is introduced and considered in the category of persistent differential graded modules. In this talk, I explain 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. This is joint work with T. Naito, S. Wakatsuki and T. Yamaguchi.
13:30 -- 14:20 Takahiro Matsushita (Shinshu University)
Homotopy types of Hom complexes whose codomains are square-free
The Hom complex Hom(G, H) of graphs is a simplicial complex constructed from a pair of graphs G and H, and its homotopy type is of interest in the graph coloring problem. Recently, Soichiro Fujii, Yuni Iwamasa, Kei Kimura, Yuta Nozaki and Akira Suzuki showed that every connected component of Hom(G, H) is aspherical if H is a cycle graph. In this talk, we show that every connected component of Hom(G, H) is aspherical if H is square-free.
14:30 -- 15:20 Shuichi Tsukuda (University of Ryukyus)
On the weak homotopy types of small finite spaces
In this talk, we show that a connected finite topological space with 12 or less points has a weak homotopy type of a wedge of spheres. In other words, we show that the order complex of a connected finite poset with 12 or less points has a homotopy type of a wedge of spheres. This is joint work with Kango Matsushima.
15:50 -- 16:40 Daisuke Kishimoto (Kyushu University)
Morse inequalities for noncompact manifolds
I will talk about Morse inequalities for a noncompact manifold with a cocompact and properly discontinuous action of a discrete group, where Morse functions are not necessarily invariant under the group action. The inequalities are given in terms of functions representing rough configurations of critical points and the L^2-Betti numbers. This is a joint work with Tsuyoshi Kato and Mitsunobu Tsutaya.
16:50 -- 17:40 Atsushi Yamaguchi (Osaka Metropolitan University)
The Steenrod algebra and its unstable representations
Let A^* be a cocommutative graded Hopf algebra over a field K. We denote by A_* the dual of A^*. By using the Milnor coaction, it can be shown that the category Mod(A^*) of left A^*-modules which are finite type and coconective is isomorphic to the category Comod(A_*) of right A_*-comodules which are finite type and coconective. If we give a filtration of A^* with certain properties, we can define a notion of unstable A^*-modules. We denote by UMod(A^*) a full subcategory of Mod(A^*) consisting unstable A^*-modules. We also define a notion of unstable A_*-comodule and denote by UComod(A_*) a full subcategory of Comod(A_*) consisting unstable A_*-comodules. We show that the isomorphism of categories between Mod(A^*) and Comod(A_*) restricts to an isomorphism of categories between UMod(A^*) and UComod(A_*) and that the inclusion functors UMod(A^*) --> Mod(A^*) and UComod(A_*) --> Comod(A_*) have a left adjoint and a right adjoint, respectively. By making use of these adjoints and the isomorphism Mod(A^*) --> Comod(A_*) of categories, we develop an "unstable" representation theory of the affine group scheme represented by A_*.
13 September 2024
Kansai Algebraic Topology Seminar
Venue
Osaka Metroplitan University, Nakamozu Campus, Building A14, 3rd Floor Room 305
Schedule
15:15 -- 16:15 Yuki Minowa (Kyoto University)
Parametrized topological complexity of spherical fibrations over spheres
Let E → B be a fibration. Then the parametrized topological complexity TC[E → B] is defined as a topological invariant of the map. It was first introduced by Cohen, Farber and Weinberger to formulate the robot motion planning problem which involves external constraints. They also gave upper and lower bounds of TC[E → B].In this talk, I will talk about the parametrized topological complexity of spherical fibrations over spheres. More precisely, I will consider a shaper lower bound given by the weak category of a certain space. Then I will show that, for n odd with n ≥ 3, there exists a fibration S^n → E → S^{m+1} such that TC[E → S^{m+1}] equals to 2. I will also talk about the determination of the parametrized topological complexity of certain bundles, especially the unit tangent bundle of even dimensional spheres.
16:30 -- 17:30 Hellen Colman (Wilbur Wright College)
Equivariant Motion Planning
Consider the space X of all possible configurations of a mechanical system. A motion planning algorithm assigns to each pair of initial and final configurations, a continuous motion of the system between them. Topological complexity is an integer TC(X) reflecting the complexity of motion planning algorithms for all systems having X as their configuration space. Roughly, TC(X) is the minimal number of continuous rules which are needed in a motion planning algorithm. This invariant was introduced by Farber in 2002 and is closely related to the classical Lusternik-Schnirelmann category. In recent years, several versions of topological complexity aimed at exploiting the presence of a group acting on the configuration space have appeared. We will present several approaches to describing equivariant topological complexity variants. In particular we will show a topological complexity suitable for orbifolds described as translation groupoids.
14 March 2024
Kansai Algebraic Topology Seminar
Venue
Nishijin plaza, Multi-purpose room
Schedule
14:00 -- 15:00 Daisuke Kishimoto (Kyushu University)
Van Kampen-Flores theorem for cell complexes
The van Kampen-Flores theorem states that the n-skeleton of a (2n+2)-simplex does not embed into R^{2n}. I will present its generalization to a continuous map from a skeleton of a certain regular CW complex. In particular, I will show that the n-skeleton of any simplicial (2n+1)-sphere does not embed into R^{2n}.
This is a joint work with Takahiro Matsushita.
15:15 -- 16:15 Yuki Minowa (Kyoto University)
A short elementary proof of Beben and Theriault's theorem on homotopy fibers
Beben and Theriault proved a theorem on the homotopy fiber of an extension of a map with respect to a cone attachment, which has produced several applications. I will talk about a short and elementary proof of this theorem.
This is a joint work with Daisuke Kishimoto.
16:30 -- 17:30 Masaki Kameko (Shibaura Institute of Technology)
Torsion in classifying spaces of gauge groups
Tsukuda showed that the integral homology of the classifying space of the gauge group of the nontrivial SO(3)-bundle over the 2-dimensional sphere has no torsion. SO(3) is isomorphic to the projective unitary group PU(2). I will generalize Tsukuda's result on the SO(3)-bundle to PU(n)-bundles. This talk is based on my recent preprint with the same title, arXiv:2401.00199.
30 January 2024
Kansai Algebraic topology Seminar
(co-organized with Shinshu Topology Seminar at Shinshu University)
Venue
Shinshu University, Science Building A, Room A-401
Schedule
14:15 -- 15:45 Atsushi Yamaguchi (Osaka Metropolitan University)
A theory of plots
The notion of plots in diffeology is introduced to define diffeological spaces which generalize differentiable manifolds. We observe that the notion of plots in diffeology has an easy generalization by replacing the site (O,E) of open sets of Euclidean spaces and open embeddings by a general Grothendieck site (C,J) and the forgetful functor U:O → Set by a set valued functor F:C → Set. In this talk, we show that the category of “generalized” plots is a quasi-topos, namely it is (finitely) complete and cocommplete, locally cartesian closed and has a strong subobject classifier. We also show that the groupoid associated with an epimorphism can be defined as in the text book “Diffeology” by P.I-Zemmour, so that we can develop a theory of fibrations in the category of “generalized” plots. Moreover, we mention the notion of F-topology which generalizes the D-topology in diffeology.
16:15 -- 17:30 Daisuke Kishimoto (Kyushu University)
Tight complexes are Golod
Tightness of a simplicial complex is a combinatorial analogue of a tight embedding of a manifold into a Euclidean space, studied in differential geometry. Golodness is a property of a noetherian ring, defined in terms of the Poincare series of its Koszul homology, and Golodness of a simplicial complex is defined by that of the Stanley-Reisner ring. Recent results on polyhedral products suggest a connection between these two notions for manifold triangulations, and Iriye and I proved that they are equivalent for 3-dimensional manifold triangulations. In this talk, I will present that tight complexes are always Golod, which implies Golodness and tightness are equivalent for all manifold triangulations. I will also give a quick survey on the study of Golodness through polyhedral products.
This is a joint work with Kouyemon Iriye.
11 October 2023
Osaka Metropolitan University, Sugimoto Campus, Building F, Room 415
16:45 -- 17:45 Norio Iwase (Kyushu University)
A closed manifold is a fat smooth CW complex
We introduce a new idea of a smooth CW complex called a “fat" smooth CW complex, which in some sense includes all CW complexes, and show that a closed manifold is a fat smooth CW compex as a “regular" smooth CW complex. To show that, we analyse the handle decomposition of a closed manifold. However, in the usual procedure of a handle decomposition, we first attach a handle on a manifold with boundary to obtain a "manifold with corners", and then, we have to smoothen the corners to obtain a manifold with boundary again. We will perform this in the category Diffeology. We also show that any topological CW complex is topologically homotopy equivalent to a “thin" smooth CW complex, most of which is not a manifold but a fat smooth CW complex.
This is a joint work with Yuki Kojima.
19 May 2023
Kansai Algebraic Topology Seminar
(co-organized with Topology Seminar at Kyushu University)
Kyushu University, Building W1, Room C514
13:40 -- 14:40 Daisuke Kishimoto (Kyushu University)
Vector fields on non-compact manifolds
Let M be a non-compact connected manifold with a cocompact and properly discontinuous action of a group G. We define the integral in the bounded de Rham cohomology of M, and establish the Hopf-Poincaré theorem for M. Then we apply it to prove that a bounded and tame vector field on M must have inifinitely many zeros whenever M/G is orientable, the Euler characteristic of M/G is non-trivial, and G is an amenable group having an element of infinite order.
This is a joint work with Tsuyoshi Kato and Mitsunobu Tsutaya.
14:50 -- 15:50 Takahiro Matsushita (University of Ryukyus)
Van Kampen-Flores theorem and Stiefel-Whitney classes
The van Kampen-Flores theorem states that the d-skeleton of a (2d+2)-simplex does not embed into R^{2d}. We prove the van Kampen-Flores theorem for triangulations of manifolds satisfying a certain condition on their Stiefel-Whitney classes. In particular, we show that the d-skeleton of a triangulation of a (2d+1)-manifold with non-trivial total Stiefel-Whitney class does not embed into R^{2d}.
This is a joint work with Daisuke Kishimoto.
17 February 2023
Kansai Algebraic Topology Seminar
Osaka Metropolitan University, Sugimoto Campus, Building E, Room 408
10:30 -- 11:30 Atsushi Yamaguchi (Osaka Metropolitan University)
Unstable modules as representations of Steenrod groups
Let G_p be an affine group scheme represented by the dual of the Steenrod algebra over a prime field of characteristic p. We call an affine group scheme G "a Steenrod group" if G is a quotient group of a subgroup of G_p. The aim of this talk is to report t he current status of my attempt to provide a foundation of a representation theory of Steenrod groups as a generalization of the theory of unstable modules over the Steenrod algebra developed by J. Lannes and others.
13:30 -- 14:30 Yichen Tong (Kyoto University)
Rational self-closeness numbers of mapping spaces
For a closed simply-connected 2n-dimensional manifold M, it has been proved that the components of the free mapping space from M to 2n-sphere have exactly two different rational homotopy types. However, since this result is proved by algebraic models of components, we do not know whether other homotopy invariants distinguish these two types or not. In this talk, we completely determine the self-closeness numbers of rationalized components of the mapping space and prove that they do distinguish different rational homotopy types. The methods also have potential to be extended to other mapping spaces.
14:45 -- 15:45 Mitsunobu Tsutaya (Kyushu University)
An associative model of homotopy coherent functors and natural transformations
Sugawara introduced homotopy coherent morphisms between topological monoids. Following his idea, we define models of homotopy coherent functors and natural transformations between topological categories. As an application, we will see that homotopy coherent morphisms between homotopy coherent algebras over an operad with associative compositions are naturally defined.
16:00 -- 17:00 Sho Hasui (Osaka Metropolitan University)
The Stiefel-Whitney classes of moment-angle manifolds are trivial
If a moment-angle complex Z_K is a smooth manifold, we can easily see that Z_K is null-cobordant (i.e. Z_K=∂M for some manifold with boundary M) and therefore the Stiefel--Whitney numbers of Z_K are trivial. This observation naturally leads us to ask the question: Are the Stiefel--Whitney classes of a moment-angle manifold also trivial? In this talk, I'd like to show that this problem can be solved affirmatively. Note that, by introducing the notion of Stiefel--Whitney classes for topological manifolds due to Fadell, we can consider this problem for the moment-angle manifolds which are topological manifolds, not necessarily smooth. For such cases, the problem is also solved affirmatively. Moreover, we also consider the quotient of a moment-angle manifold by a subtorus acting freely on it. If the dimension of the subtorus is less than a canonical upper bound, then the Stiefel--Whitney numbers of the quotient manifold are trivial.