Kansai Algebraic Topology Seminar
Description
This seminar aims at exchanging recent ideas and techniques around algebraic topology through lectures and discussions.
Organizers
Sho Hasui (Osaka Metropolitan University)
Atsushi Yamaguchi (Osaka Metropolitan University)
Upcoming Seminars
To be announced
Past Seminars
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.
30th January 2024
(co-organized by 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.
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.
19th May 2023
(co-organized by 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.
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.