AlgebraiC Topology NETWORK
Description
Algebraic Topology Network, formerly known as Kansai Algebraic Seminar, organizes seminars and workshops 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)
Upcoming Event
29th January 2025
One Day Workshop with Rachael Boyd on Algebraic Topology
Organizer
Tsuyoshi Kato
Venue
Kyoto University, Sceience Bldg 3, Room 127
Schedule
9:30 -- 10:30 Tadayuki Watanabe
10:45 --11:45 Daisuke Kishimoto
Lunch
13:30 -- 14:30 Kensuke Arakawa
14:45 -- 15:45 Toshiyuki Akita
16:00 -- 17:00 Rachael Boyd
Abstract
-- Tadayuki Watanabe
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.
-- Daisuke Kishimoto
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.
-- Kensuke Arakawa
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.
-- Toshiyuki Akita
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.
-- Rachael Boyd
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.
Past Events
7th Meeting 9th 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_*.
6th Meeting 13th September 2024
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.
5th Meeting 14th March 2024
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.
4th Meeting 30th January 2024
(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.
3rd Meeting 11th 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.
2nd Meeting 19th May 2023
(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.
1st Meeting 17th February 2023
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.