## Tuesday Seminar on Topology

Seminar information archive ～09/23｜Next seminar｜Future seminars 09/24～

Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |

**Seminar information archive**

### 2022/06/14

17:30-18:30 Online

Pre-registration required. See our seminar webpage.

Cartan calculi on the free loop spaces (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Katsuhiko Kuribayashi**(Shinshu University)Cartan calculi on the free loop spaces (JAPANESE)

[ Abstract ]

A typical example of a Cartan calculus is the Lie algebra representation of vector fields of a manifold on the derivation ring of the de Rham complex. In this talk, a `second stage' of the Cartan calculus is investigated. In a more general setting, the stage is formulated with a Lie algebra representation of the Andre-Quillen cohomology of a commutative differential graded algebra A on the endomorphism ring of the Hochschild homology of A in terms of the homotopy Cartan calculi in the sense of Fiorenza and Kowalzig. Moreover, the Lie algebra representation in the Cartan calculus is interpreted geometrically as a map from the rational homotopy group of the monoid of self-homotopy equivalences on a simply-connected space M to the derivation ring on the loop cohomology of M. An extension of the representation to the string cohomology and its geometric counterpart are also discussed together with the BV exactness which is a new rational homotopy invariant introduced in our work. This talk is based on joint work in progress with T. Naito, S. Wakatsuki and T. Yamaguchi.

[ Reference URL ]A typical example of a Cartan calculus is the Lie algebra representation of vector fields of a manifold on the derivation ring of the de Rham complex. In this talk, a `second stage' of the Cartan calculus is investigated. In a more general setting, the stage is formulated with a Lie algebra representation of the Andre-Quillen cohomology of a commutative differential graded algebra A on the endomorphism ring of the Hochschild homology of A in terms of the homotopy Cartan calculi in the sense of Fiorenza and Kowalzig. Moreover, the Lie algebra representation in the Cartan calculus is interpreted geometrically as a map from the rational homotopy group of the monoid of self-homotopy equivalences on a simply-connected space M to the derivation ring on the loop cohomology of M. An extension of the representation to the string cohomology and its geometric counterpart are also discussed together with the BV exactness which is a new rational homotopy invariant introduced in our work. This talk is based on joint work in progress with T. Naito, S. Wakatsuki and T. Yamaguchi.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/06/07

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**早稲田大学**(Waseda University)Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups (JAPANESE)

[ Abstract ]

We discuss a relation between a dynamical zeta function defined by the geodesic flow on a 2-dimensional hyperbolic orbifold and the asymptotic behavior of the Reidemeister torsion for the unit tangent bundle over the orbifold. The unit tangent bundle over a hyperbolic orbifold is a Seifert fibered space with a geometric structure given by the universal cover of PSL(2, R). This geometric structure induces an SL(2, R)-representation of the fundamental group. Here the asymptotic behavior of the Reidemeister torsion means the limit of the leading coefficient in the Reidemeister torsion for the unit tangent bundle over a hyperbolic orbifold and the SL(n, R)-representations induced by the SL(2, R)-one of its fundamental group. For a hyperbolic 3-manifold, we can derive the hyperbolic volume from the limit of the leading coefficient in the Reidemeister torsion with a dynamical zeta function according to previous works. For the unit tangent bundle over a 2-dimensional hyperbolic orbifold, which is not a hyperbolic 3-manifold, we can find the orbifold Euler characteristic of the orbifold in the limit of the leading coefficient in the Reidemeister torsion for the unit tangent bundle from the relation with the dynamical zeta function defined by the geodesic flow on the orbifold.

[ Reference URL ]We discuss a relation between a dynamical zeta function defined by the geodesic flow on a 2-dimensional hyperbolic orbifold and the asymptotic behavior of the Reidemeister torsion for the unit tangent bundle over the orbifold. The unit tangent bundle over a hyperbolic orbifold is a Seifert fibered space with a geometric structure given by the universal cover of PSL(2, R). This geometric structure induces an SL(2, R)-representation of the fundamental group. Here the asymptotic behavior of the Reidemeister torsion means the limit of the leading coefficient in the Reidemeister torsion for the unit tangent bundle over a hyperbolic orbifold and the SL(n, R)-representations induced by the SL(2, R)-one of its fundamental group. For a hyperbolic 3-manifold, we can derive the hyperbolic volume from the limit of the leading coefficient in the Reidemeister torsion with a dynamical zeta function according to previous works. For the unit tangent bundle over a 2-dimensional hyperbolic orbifold, which is not a hyperbolic 3-manifold, we can find the orbifold Euler characteristic of the orbifold in the limit of the leading coefficient in the Reidemeister torsion for the unit tangent bundle from the relation with the dynamical zeta function defined by the geodesic flow on the orbifold.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/05/31

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Stable Fukaya categories of Milnor fibers (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Kazushi Ueda**(The Univesity of Tokyo)Stable Fukaya categories of Milnor fibers (JAPANESE)

[ Abstract ]

We define the stable Fukaya category of a Liouville domain as the quotient of the wrapped Fukaya category by the full subcategory consisting of compact Lagrangians, and discuss the relation between the stable Fukaya categories of affine Fermat hypersurfaces and the Fukaya categories of projective hypersurfaces. We also discuss homological mirror symmetry for Milnor fibers of Brieskorn-Pham singularities along the way. This is a joint work in progress with Yanki Lekili.

[ Reference URL ]We define the stable Fukaya category of a Liouville domain as the quotient of the wrapped Fukaya category by the full subcategory consisting of compact Lagrangians, and discuss the relation between the stable Fukaya categories of affine Fermat hypersurfaces and the Fukaya categories of projective hypersurfaces. We also discuss homological mirror symmetry for Milnor fibers of Brieskorn-Pham singularities along the way. This is a joint work in progress with Yanki Lekili.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/05/24

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Polynomial functors associated with beaded open Jacobi diagrams (ENGLISH)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Christine Vespa**(IRMA, Université de Strasbourg / JSPS)Polynomial functors associated with beaded open Jacobi diagrams (ENGLISH)

[ Abstract ]

The Kontsevich integral is a very powerful invariant of knots, taking values is the space of Jacobi diagrams. Using an extension of the Kontsevich integral to tangles in handlebodies, Habiro and Massuyeau construct a functor from the category of bottom tangles in handlebodies to the linear category A of Jacobi diagrams in handlebodies. The category A has a subcategory equivalent to the linearization of the opposite of the category of finitely generated free groups, denoted by $\textbf{gr}^{op}$. By restriction to this subcategory, morphisms in the linear category $\textbf{A}$ give rise to interesting contravariant functors on the category $\textbf{gr}$, encoding part of the composition structure of the category A.

In recent papers, Katada studies the functor given by the morphisms in the category A from 0. In particular, she obtains a family of polynomial functors on $\textbf{gr}^{op}$ which are outer functors, in the sense that inner automorphisms act trivially.

In this talk, I will explain these results and give extensions of Katada’s results concerning the functors given by the morphisms in the category A from any integer k. These functors give rise to families of polynomial functors on $\textbf{gr}^{op}$ which are no more outer functors. Our approach is based on an equivalence of categories given by Powell. Through this equivalence the previous polynomial functors correspond to functors given by beaded open Jacobi diagrams.

[ Reference URL ]The Kontsevich integral is a very powerful invariant of knots, taking values is the space of Jacobi diagrams. Using an extension of the Kontsevich integral to tangles in handlebodies, Habiro and Massuyeau construct a functor from the category of bottom tangles in handlebodies to the linear category A of Jacobi diagrams in handlebodies. The category A has a subcategory equivalent to the linearization of the opposite of the category of finitely generated free groups, denoted by $\textbf{gr}^{op}$. By restriction to this subcategory, morphisms in the linear category $\textbf{A}$ give rise to interesting contravariant functors on the category $\textbf{gr}$, encoding part of the composition structure of the category A.

In recent papers, Katada studies the functor given by the morphisms in the category A from 0. In particular, she obtains a family of polynomial functors on $\textbf{gr}^{op}$ which are outer functors, in the sense that inner automorphisms act trivially.

In this talk, I will explain these results and give extensions of Katada’s results concerning the functors given by the morphisms in the category A from any integer k. These functors give rise to families of polynomial functors on $\textbf{gr}^{op}$ which are no more outer functors. Our approach is based on an equivalence of categories given by Powell. Through this equivalence the previous polynomial functors correspond to functors given by beaded open Jacobi diagrams.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/05/17

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Contribution of simple loops to the configuration space integral (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Tatsuro Shimizu**(Tokyo Denki University)Contribution of simple loops to the configuration space integral (JAPANESE)

[ Abstract ]

For a manifold with a representation of the fundamental group, the configuration space integral associated with a graph (Feynman diagram) gives a real number. An appropriate linear combination of graphs gives an invariant of the manifold with the representation. In this talk, we discuss the contribution of simple loops to the configuration space integral. Hutchings, Lee and Kitayama give geometric descriptions of the Reidemeister torsion by using circle valued Morse functions. By using these descriptions and a Morse theoretical description of the configuration space integral, we have equations among the Reidemeister torsion and the contributions of simple loops in some cases. In this talk, we extend the equations for some other cases and give a computational example of the configuration space integrals by using Morse function.

[ Reference URL ]For a manifold with a representation of the fundamental group, the configuration space integral associated with a graph (Feynman diagram) gives a real number. An appropriate linear combination of graphs gives an invariant of the manifold with the representation. In this talk, we discuss the contribution of simple loops to the configuration space integral. Hutchings, Lee and Kitayama give geometric descriptions of the Reidemeister torsion by using circle valued Morse functions. By using these descriptions and a Morse theoretical description of the configuration space integral, we have equations among the Reidemeister torsion and the contributions of simple loops in some cases. In this talk, we extend the equations for some other cases and give a computational example of the configuration space integrals by using Morse function.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/05/10

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Nielsen realization, knots, and Seiberg-Witten (Floer) homotopy theory (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Hokuto Konno**(The Univesity of Tokyo)Nielsen realization, knots, and Seiberg-Witten (Floer) homotopy theory (JAPANESE)

[ Abstract ]

I will discuss two different kinds of applications of Seiberg-Witten (Floer) homotopy theory involving involutions. The first application is about the Nielsen realization problem, which asks whether a given finite subgroup of the mapping class group of a manifold lifts to a subgroup of the diffeomorphism group. Although every finite subgroup is known to lift in dimension 2, there are manifolds of dimension greater than 2 for which the Nielsen realization fails. However, only few examples have been known in dimension 4. I will show that "4-dimensional Dehn twists" yield a large class of new examples. The second application is about 4-dimensional invariants of knots. I will introduce a version of "Floer K-theory for knots", and will explain that this framework gives the first comparison result for the smooth and topological versions of a certain knot invariant, called stabilizing number. Although the above two topics (Nielsen realization and knots) may seem to have different flavors, they are derived from a common idea. The first one is proved using a constraint on smooth involutions on a closed 4-manifold from Seiberg-Witten homotopy theory by Yuya Kato, and the second one is derived from a generalization of Kato's result to 4-manifolds with boundary using Seiberg-Witten Floer homotopy theory. This talk is partially based on joint work with Jin Miyazawa and Masaki Taniguchi.

[ Reference URL ]I will discuss two different kinds of applications of Seiberg-Witten (Floer) homotopy theory involving involutions. The first application is about the Nielsen realization problem, which asks whether a given finite subgroup of the mapping class group of a manifold lifts to a subgroup of the diffeomorphism group. Although every finite subgroup is known to lift in dimension 2, there are manifolds of dimension greater than 2 for which the Nielsen realization fails. However, only few examples have been known in dimension 4. I will show that "4-dimensional Dehn twists" yield a large class of new examples. The second application is about 4-dimensional invariants of knots. I will introduce a version of "Floer K-theory for knots", and will explain that this framework gives the first comparison result for the smooth and topological versions of a certain knot invariant, called stabilizing number. Although the above two topics (Nielsen realization and knots) may seem to have different flavors, they are derived from a common idea. The first one is proved using a constraint on smooth involutions on a closed 4-manifold from Seiberg-Witten homotopy theory by Yuya Kato, and the second one is derived from a generalization of Kato's result to 4-manifolds with boundary using Seiberg-Witten Floer homotopy theory. This talk is partially based on joint work with Jin Miyazawa and Masaki Taniguchi.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/04/26

17:00-18:00 Online

Joint with Lie Groups and Representation Theory Seminar. See our seminar webpage.

On the existence of discrete series for homogeneous spaces (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Joint with Lie Groups and Representation Theory Seminar. See our seminar webpage.

**Yoshiki Oshima**(The Univesity of Tokyo)On the existence of discrete series for homogeneous spaces (JAPANESE)

[ Abstract ]

When a Lie group $G$ acts transitively on a manifold $X$, an irreducible subrepresentation of $L^2(X)$ is called a discrete series representation of $X$. One may ask which homogeneous space $X$ has a discrete series representation. For reductive symmetric spaces, it is known that the existence of discrete series is equivalent to a rank condition by works of Flensted-Jensen, T.Matsuki, and T.Oshima. The problem for general reductive homogeneous spaces was considered by T.Kobayashi and a sufficient condition for the existence of discrete series was obtained by using his theory of admissible restriction. In this talk, we would like to see another sufficient condition for general homogeneous spaces and also the case of their line bundles in terms of the orbit method.

[ Reference URL ]When a Lie group $G$ acts transitively on a manifold $X$, an irreducible subrepresentation of $L^2(X)$ is called a discrete series representation of $X$. One may ask which homogeneous space $X$ has a discrete series representation. For reductive symmetric spaces, it is known that the existence of discrete series is equivalent to a rank condition by works of Flensted-Jensen, T.Matsuki, and T.Oshima. The problem for general reductive homogeneous spaces was considered by T.Kobayashi and a sufficient condition for the existence of discrete series was obtained by using his theory of admissible restriction. In this talk, we would like to see another sufficient condition for general homogeneous spaces and also the case of their line bundles in terms of the orbit method.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/04/19

17:30-18:30 Online

Joint with Lie Groups and Representation Theory Seminar. See our seminar webpage.

On the classification and construction of conformal symmetry breaking operators for anti-de Sitter spaces (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Joint with Lie Groups and Representation Theory Seminar. See our seminar webpage.

**Toshihisa Kubo**(Ryukoku University)On the classification and construction of conformal symmetry breaking operators for anti-de Sitter spaces (JAPANESE)

[ Abstract ]

Let $X$ be a smooth manifold and $Y$ a smooth submanifold of $X$. Take $G' \subset G$ to be a pair of Lie groups that act on $Y \subset X$, respectively. Consider a $G'$-intertwining differential operator $\mathcal{D}$ from the space of smooth sections for a $G$-equivariant vector bundle over $X$ to that for a $G'$-equivariant vector bundle over $Y$. Toshiyuki Kobayashi called such a differential operator $\mathcal{D}$ a

In [Kobayashi-K-Pevzner, Lecture Notes in Math. 2170 (2016)], we explicitly constructed and classified all the differential SBOs from the space of differential $i$-forms $\mathcal{E}^i(S^n)$ over the standard Riemann sphere $S^n$ to that of differential $j$-forms $\mathcal{E}^j(S^{n-1})$ over the totally geodesic hypersphere $S^{n-1}$. In this talk, by extending the results in a Riemannian setting, we discuss about the classification and construction of differential SBOs in a pseudo-Riemannian setting such as anti-de Sitter spaces and hyperbolic spaces. This is a joint work with Toshiyuki Kobayashi and Michael Pevzner.

[ Reference URL ]Let $X$ be a smooth manifold and $Y$ a smooth submanifold of $X$. Take $G' \subset G$ to be a pair of Lie groups that act on $Y \subset X$, respectively. Consider a $G'$-intertwining differential operator $\mathcal{D}$ from the space of smooth sections for a $G$-equivariant vector bundle over $X$ to that for a $G'$-equivariant vector bundle over $Y$. Toshiyuki Kobayashi called such a differential operator $\mathcal{D}$ a

**differential symmetry breaking operator**(differential SBO for short) ([T.~Kobayashi, Differential Geom. Appl. (2014)]).In [Kobayashi-K-Pevzner, Lecture Notes in Math. 2170 (2016)], we explicitly constructed and classified all the differential SBOs from the space of differential $i$-forms $\mathcal{E}^i(S^n)$ over the standard Riemann sphere $S^n$ to that of differential $j$-forms $\mathcal{E}^j(S^{n-1})$ over the totally geodesic hypersphere $S^{n-1}$. In this talk, by extending the results in a Riemannian setting, we discuss about the classification and construction of differential SBOs in a pseudo-Riemannian setting such as anti-de Sitter spaces and hyperbolic spaces. This is a joint work with Toshiyuki Kobayashi and Michael Pevzner.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/01/25

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Some obstructions on subgroups of the Brin-Thompson group $2V$ (ENGLISH)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Xiaobing Sheng**(The Univesity of Tokyo)Some obstructions on subgroups of the Brin-Thompson group $2V$ (ENGLISH)

[ Abstract ]

Motivated by Burillo, Cleary and Röver's summary of the obstruction for subgroups of Thompson's group $V$, we investigate the higher dimensional version, the group $2V$ and found out that they have similar obstructions on torsion subgroups and certain Baumslag-Solitar groups.

[ Reference URL ]Motivated by Burillo, Cleary and Röver's summary of the obstruction for subgroups of Thompson's group $V$, we investigate the higher dimensional version, the group $2V$ and found out that they have similar obstructions on torsion subgroups and certain Baumslag-Solitar groups.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2022/01/11

17:00-18:00 Online

Joint with Lie Groups and Representation Theory Seminar. See our seminar webpage.

On the existence problem of Compact Clifford-Klein forms of indecomposable pseudo-Riemannian symmetric spaces with signature (n,2) (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Joint with Lie Groups and Representation Theory Seminar. See our seminar webpage.

**Keiichi Maeta**(The Univesity of Tokyo)On the existence problem of Compact Clifford-Klein forms of indecomposable pseudo-Riemannian symmetric spaces with signature (n,2) (JAPANESE)

[ Abstract ]

For a homogeneous space $G/H$ and its discontinuous group $\Gamma\subset G$, the double coset space $\Gamma\backslash G/H$ is called a Clifford-Klein form of $G/H$. In the study of Clifford-Klein forms, the classification of homogeneous spaces which admit compact Clifford—Klein forms is one of the important open problems, which was introduced by Toshiyuki Kobayashi in 1980s. We consider this problem for indecomposable and reducible pseudo-Riemannian symmetric spaces with signature (n,2). We show the non-existence of compact Clifford-Klein forms for some series of symmetric spaces, and construct new compact Clifford-Klein forms of countably infinite five-dimensional pseudo-Riemannian symmetric spaces with signature (3,2).

[ Reference URL ]For a homogeneous space $G/H$ and its discontinuous group $\Gamma\subset G$, the double coset space $\Gamma\backslash G/H$ is called a Clifford-Klein form of $G/H$. In the study of Clifford-Klein forms, the classification of homogeneous spaces which admit compact Clifford—Klein forms is one of the important open problems, which was introduced by Toshiyuki Kobayashi in 1980s. We consider this problem for indecomposable and reducible pseudo-Riemannian symmetric spaces with signature (n,2). We show the non-existence of compact Clifford-Klein forms for some series of symmetric spaces, and construct new compact Clifford-Klein forms of countably infinite five-dimensional pseudo-Riemannian symmetric spaces with signature (3,2).

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/12/21

17:30-18:30 Online

Joint with Lie Groups and Representation Theory Seminar. See our seminar webpage.

Classification of holomorphic vertex operator algebras of central charge 24 (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Joint with Lie Groups and Representation Theory Seminar. See our seminar webpage.

**Hiroki Shimakura**(Tohoku University)Classification of holomorphic vertex operator algebras of central charge 24 (JAPANESE)

[ Abstract ]

Holomorphic vertex operator algebras are imporant in vertex operator algebra theory. For example, the famous moonshine vertex operator algebra is holomorphic. One of the fundamental problems is to classify holomorphic vertex operator algebras. It is known that holomorphic vertex operator algebras of central charge 8 and 16 are lattice vertex operator algebras. I will talk about recent progress on the classification of holomorphic vertex operator algebras of central charge 24.

[ Reference URL ]Holomorphic vertex operator algebras are imporant in vertex operator algebra theory. For example, the famous moonshine vertex operator algebra is holomorphic. One of the fundamental problems is to classify holomorphic vertex operator algebras. It is known that holomorphic vertex operator algebras of central charge 8 and 16 are lattice vertex operator algebras. I will talk about recent progress on the classification of holomorphic vertex operator algebras of central charge 24.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/12/07

17:00-1800 Online

Pre-registration required. See our seminar webpage.

A Bar-Natan homotopy type (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Taketo Sano**(The Univesity of Tokyo)A Bar-Natan homotopy type (JAPANESE)

[ Abstract ]

In year 2000, Khovanov introduced a categorification of the Jones polynomial, now known as Khovanov homology. In 2014, Lipshitz and Sarkar introduced a spatial refinement of Khovanov homology, called Khovanov homotopy type, which is a finite CW spectrum whose reduced cellular cohomology recovers Khovanov homology. On the algebraic level, there are several deformations of Khovanov homology, such as Lee homology and Bar-Natan homology. These variants are also important in that they give knot invariants such as Rasmussen’s $s$-invariant. Whether these variants admit spatial refinements have been open.

In 2021, the speaker constructed a spatial refinement of Bar-Natan homology and determined its stable homotopy type. The construction follows that of Lipshitz and Sarkar, which is based on the construction proposed by Cohen, Segal and Jones using the concept of flow categories. Also, we adopt techniques called “Morse moves in flow categories” introduced by Lobb et.al. to determine the stable homotopy type. Spacialy (or homotopically) refining the $s$-invariant is left as a future work.

[ Reference URL ]In year 2000, Khovanov introduced a categorification of the Jones polynomial, now known as Khovanov homology. In 2014, Lipshitz and Sarkar introduced a spatial refinement of Khovanov homology, called Khovanov homotopy type, which is a finite CW spectrum whose reduced cellular cohomology recovers Khovanov homology. On the algebraic level, there are several deformations of Khovanov homology, such as Lee homology and Bar-Natan homology. These variants are also important in that they give knot invariants such as Rasmussen’s $s$-invariant. Whether these variants admit spatial refinements have been open.

In 2021, the speaker constructed a spatial refinement of Bar-Natan homology and determined its stable homotopy type. The construction follows that of Lipshitz and Sarkar, which is based on the construction proposed by Cohen, Segal and Jones using the concept of flow categories. Also, we adopt techniques called “Morse moves in flow categories” introduced by Lobb et.al. to determine the stable homotopy type. Spacialy (or homotopically) refining the $s$-invariant is left as a future work.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/11/30

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

A non-commutative Reidemeister-Turaev torsion of homology cylinders (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Masatoshi Sato**(Tokyo Denki University)A non-commutative Reidemeister-Turaev torsion of homology cylinders (JAPANESE)

[ Abstract ]

The Reidemeister-Turaev torsion of homology cylinders takes values in the integral group ring of the first homology of a surface. We lift it to a torsion valued in the $K_1$-group of the completed rational group ring of the fundamental group of the surface. We show that it induces a finite type invariant of homology cylinders, and describe the induced map on the graded quotient of the Y-filtration of homology cylinders via the 1-loop part of the LMO functor and the Enomoto-Satoh trace. This talk is based on joint work with Yuta Nozaki and Masaaki Suzuki.

[ Reference URL ]The Reidemeister-Turaev torsion of homology cylinders takes values in the integral group ring of the first homology of a surface. We lift it to a torsion valued in the $K_1$-group of the completed rational group ring of the fundamental group of the surface. We show that it induces a finite type invariant of homology cylinders, and describe the induced map on the graded quotient of the Y-filtration of homology cylinders via the 1-loop part of the LMO functor and the Enomoto-Satoh trace. This talk is based on joint work with Yuta Nozaki and Masaaki Suzuki.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/11/16

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Skein and cluster algebras of marked surfaces without punctures for sl(3) (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Wataru Yuasa**(RIMS, Kyoto University)Skein and cluster algebras of marked surfaces without punctures for sl(3) (JAPANESE)

[ Abstract ]

We consider a skein algebra consisting of sl(3)-webs with the boundary skein relations for a marked surface without punctures. We construct a quantum cluster algebra coming from the moduli space of decorated SL(3)-local systems of the surface inside the skew-field of fractions of the skein algebra. In this talk, we introduce the sticking trick and the cutting trick for sl(3)-webs. The sticking trick expands the boundary-localized skein algebra into the cluster algebra. The cutting trick gives Laurent expressions of "elevation-preserving" webs with positive coefficients in certain clusters. We can also apply these tricks in the case of sp(4). This talk is based on joint works with Tsukasa Ishibashi.

[ Reference URL ]We consider a skein algebra consisting of sl(3)-webs with the boundary skein relations for a marked surface without punctures. We construct a quantum cluster algebra coming from the moduli space of decorated SL(3)-local systems of the surface inside the skew-field of fractions of the skein algebra. In this talk, we introduce the sticking trick and the cutting trick for sl(3)-webs. The sticking trick expands the boundary-localized skein algebra into the cluster algebra. The cutting trick gives Laurent expressions of "elevation-preserving" webs with positive coefficients in certain clusters. We can also apply these tricks in the case of sp(4). This talk is based on joint works with Tsukasa Ishibashi.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/11/09

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

The spaces of non-descendible quasimorphisms and bounded characteristic classes (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Shuhei Maruyama**(Nagoya University)The spaces of non-descendible quasimorphisms and bounded characteristic classes (JAPANESE)

[ Abstract ]

A quasimorphism is a real-valued function on a group which is a homomorphism up to bounded error. In this talk, we discuss the (non-)descendibility of quasimorphisms. In particular, we consider the space of non-descendible quasimorphisms on universal covering groups and explain its relation to the space of bounded characteristic classes of foliated bundles. This talk is based on a joint work with Morimichi Kawasaki.

[ Reference URL ]A quasimorphism is a real-valued function on a group which is a homomorphism up to bounded error. In this talk, we discuss the (non-)descendibility of quasimorphisms. In particular, we consider the space of non-descendible quasimorphisms on universal covering groups and explain its relation to the space of bounded characteristic classes of foliated bundles. This talk is based on a joint work with Morimichi Kawasaki.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/11/02

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Meta-nilpotent knot invariants and symplectic automorphism groups of free nilpotent groups (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Takefumi Nosaka**(Tokyo Institute of Technolog)Meta-nilpotent knot invariants and symplectic automorphism groups of free nilpotent groups (JAPANESE)

[ Abstract ]

There are many developments of fibered knots and homology cylinders from topological and algebraic viewpoints. In a converse sense, we discuss meta-nilpotent localization of knot groups,

which can deal with any knot like fibered knots. The monodoromy can be regarded as a symplectic automorphism of free nilpotent group, and the conjugacy classes in the outer automorphism groups produce knot invariants in terms of Johnson homomorphisms. In this talk, I show the construction of the monodoromies, and some results on the knot invariants. I also talk approaches from Fox pairings.

[ Reference URL ]There are many developments of fibered knots and homology cylinders from topological and algebraic viewpoints. In a converse sense, we discuss meta-nilpotent localization of knot groups,

which can deal with any knot like fibered knots. The monodoromy can be regarded as a symplectic automorphism of free nilpotent group, and the conjugacy classes in the outer automorphism groups produce knot invariants in terms of Johnson homomorphisms. In this talk, I show the construction of the monodoromies, and some results on the knot invariants. I also talk approaches from Fox pairings.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/10/26

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

On the strongly pseudoconcave boundary of a compact complex surface (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Naohiko Kasuya**(Hokkaido University)On the strongly pseudoconcave boundary of a compact complex surface (JAPANESE)

[ Abstract ]

On the strongly pseudoconvex (resp. pseudoconcave) boundary of a complex surface, the complex

tangency defines a positive (resp. negative) contact structure. Bogomolov and De Oliveira proved

that the boundary contact structure of a strongly pseudoconvex surface is Stein fillable.

Therefore, for a closed contact 3-manifold, Stein fillability and holomorphic fillability are

equivalent. Then what about the boundary of a strongly pseudoconcave surface? We prove that any

closed negative contact 3-manifold can be realized as the boundary of a strongly pseudoconcave

surface. The proof is done by establishing holomorphic handle attaching method to the strongly

pseudoconcave boundary of a complex surface, based on Eliashberg's handlebody construction of Stein

manifolds. This is a joint work with Daniele Zuddas (University of Trieste).

[ Reference URL ]On the strongly pseudoconvex (resp. pseudoconcave) boundary of a complex surface, the complex

tangency defines a positive (resp. negative) contact structure. Bogomolov and De Oliveira proved

that the boundary contact structure of a strongly pseudoconvex surface is Stein fillable.

Therefore, for a closed contact 3-manifold, Stein fillability and holomorphic fillability are

equivalent. Then what about the boundary of a strongly pseudoconcave surface? We prove that any

closed negative contact 3-manifold can be realized as the boundary of a strongly pseudoconcave

surface. The proof is done by establishing holomorphic handle attaching method to the strongly

pseudoconcave boundary of a complex surface, based on Eliashberg's handlebody construction of Stein

manifolds. This is a joint work with Daniele Zuddas (University of Trieste).

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/10/19

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Period matrices of some hyperelliptic Riemann surfaces (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Yoshihiko Shinomiya**(Shizuoka University)Period matrices of some hyperelliptic Riemann surfaces (JAPANESE)

[ Abstract ]

In this talk, we give new examples of period matrices of hyperelliptic Riemann surfaces. For generic genus, there were few examples of period matrices. The period matrix of a Riemann surface depends only on the choice of symplectic basis of the first homology group. It is difficult to find a symplectic basis in general. We construct hyperelliptic Riemann surfaces of generic genus from some rectangles and find their symplectic bases. Moreover, we give their algebraic equations. The algebraic equations are of the form $w^2=z(z^2-1)(z^2-a_1^2)(z^2-a_2^2) \cdots (z^2-a_{g-1}^2)$ ($1 < a_1< a_2< \cdots < a_{g-1}$). From them, we can calculate period matrices of our Riemann surfaces. We also show that all algebraic curves of this types of equations are obtained by our construction.

[ Reference URL ]In this talk, we give new examples of period matrices of hyperelliptic Riemann surfaces. For generic genus, there were few examples of period matrices. The period matrix of a Riemann surface depends only on the choice of symplectic basis of the first homology group. It is difficult to find a symplectic basis in general. We construct hyperelliptic Riemann surfaces of generic genus from some rectangles and find their symplectic bases. Moreover, we give their algebraic equations. The algebraic equations are of the form $w^2=z(z^2-1)(z^2-a_1^2)(z^2-a_2^2) \cdots (z^2-a_{g-1}^2)$ ($1 < a_1< a_2< \cdots < a_{g-1}$). From them, we can calculate period matrices of our Riemann surfaces. We also show that all algebraic curves of this types of equations are obtained by our construction.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/10/12

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Seiberg-Witten Floer homotopy and contact structures (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Nobuo Iida**(The Univesity of Tokyo)Seiberg-Witten Floer homotopy and contact structures (JAPANESE)

[ Abstract ]

Seiberg-Witten theory has been an efficient tool to study 4-dimensional symplectic and 3-dimensional contact geometry. In this talk, we introduce new homotopical invariants related to these structures using Seiberg-Witten theory and explain their properties and applications. These invariants have two main origins:

1. Kronheimer-Mrowka's invariant for 4-manifold with contact boundary, whose construction is based on Seiberg-Witten equation on 4-manifolds with conical end.

2. Bauer-Furuta and Manolescu's homotopical method called finite dimensional approximation in Seiberg-Witten theory.

This talk includes joint works with Masaki Taniguchi(RIKEN) and Anubhav Mukherjee(Georgia tech).

[ Reference URL ]Seiberg-Witten theory has been an efficient tool to study 4-dimensional symplectic and 3-dimensional contact geometry. In this talk, we introduce new homotopical invariants related to these structures using Seiberg-Witten theory and explain their properties and applications. These invariants have two main origins:

1. Kronheimer-Mrowka's invariant for 4-manifold with contact boundary, whose construction is based on Seiberg-Witten equation on 4-manifolds with conical end.

2. Bauer-Furuta and Manolescu's homotopical method called finite dimensional approximation in Seiberg-Witten theory.

This talk includes joint works with Masaki Taniguchi(RIKEN) and Anubhav Mukherjee(Georgia tech).

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/10/05

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Twisted Alexander polynomials, chirality, and local deformations of hyperbolic 3-cone-manifolds (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Hiroshi Goda**(Tokyo University of Agriculture and Technology)Twisted Alexander polynomials, chirality, and local deformations of hyperbolic 3-cone-manifolds (JAPANESE)

[ Abstract ]

We discuss a relationship between the chirality of knots and higher dimensional twisted Alexander polynomials associated with holonomy representations of hyperbolic $3$-cone-manifolds. In particular, we provide a new necessary condition for a knot, that appears in a hyperbolic $3$-cone-manifold of finite volume as a singular set, to be amphicheiral. Moreover, we can detect the chirality of hyperbolic twist knots, according to our criterion, using low-dimensional irreducible representations. (This is a joint work with Takayuki Morifuji.)

[ Reference URL ]We discuss a relationship between the chirality of knots and higher dimensional twisted Alexander polynomials associated with holonomy representations of hyperbolic $3$-cone-manifolds. In particular, we provide a new necessary condition for a knot, that appears in a hyperbolic $3$-cone-manifold of finite volume as a singular set, to be amphicheiral. Moreover, we can detect the chirality of hyperbolic twist knots, according to our criterion, using low-dimensional irreducible representations. (This is a joint work with Takayuki Morifuji.)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/07/13

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Homotopy motions of surfaces in 3-manifolds (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Makoto Sakuma**(Osaka City University Advanced Mathematical Institute)Homotopy motions of surfaces in 3-manifolds (JAPANESE)

[ Abstract ]

We introduce the concept of a homotopy motion of a subset in a manifold, and give a systematic study of homotopy motions of surfaces in closed orientable 3-manifolds. This notion arises from various natural problems in 3-manifold theory such as domination of manifold pairs, homotopical behaviour of simple loops on a Heegaard surface, and monodromies of virtual branched covering surface bundles associated to a Heegaard splitting. This is a joint work with Yuya Koda (arXiv:2011.05766).

[ Reference URL ]We introduce the concept of a homotopy motion of a subset in a manifold, and give a systematic study of homotopy motions of surfaces in closed orientable 3-manifolds. This notion arises from various natural problems in 3-manifold theory such as domination of manifold pairs, homotopical behaviour of simple loops on a Heegaard surface, and monodromies of virtual branched covering surface bundles associated to a Heegaard splitting. This is a joint work with Yuya Koda (arXiv:2011.05766).

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/07/06

17:30-18:30 Online

Pre-registration required. See our seminar webpage.

Codimension 2 transfer map in higher index theory (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Yosuke Kubota**(Shinshu University)Codimension 2 transfer map in higher index theory (JAPANESE)

[ Abstract ]

The Rosenberg index is a topological invariant taking value in the K-group of the C*-algebra of the fundamental group, which is a strong obstruction for a closed spin manifold to admit a positive scalar curvature (psc) metric. In 2015 Hanke-Pape-Schick proves that, for a nice codimension 2 submanifold N of M, the Rosenberg index of N obstructs to a psc metric on M. This is a far reaching generalization of a classical result of Gromov and Lawson. In this talk I introduce a joint work with T. Schick and its continuation concerned with this `codimension 2 index' obstruction. We construct a map between C*-algebra K-groups, which we call the codimension 2 transfer map, relating the Rosenberg index of M to that of N directly. This shows that Hanke-Pape-Schick's obstruction is dominated by a standard one, the Rosenberg index of M. We also extend our codimension 2 transfer map to secondary index invariants called the higher rho invariant. As a consequence, we obtain some example of psc manifolds are not psc null-cobordant.

[ Reference URL ]The Rosenberg index is a topological invariant taking value in the K-group of the C*-algebra of the fundamental group, which is a strong obstruction for a closed spin manifold to admit a positive scalar curvature (psc) metric. In 2015 Hanke-Pape-Schick proves that, for a nice codimension 2 submanifold N of M, the Rosenberg index of N obstructs to a psc metric on M. This is a far reaching generalization of a classical result of Gromov and Lawson. In this talk I introduce a joint work with T. Schick and its continuation concerned with this `codimension 2 index' obstruction. We construct a map between C*-algebra K-groups, which we call the codimension 2 transfer map, relating the Rosenberg index of M to that of N directly. This shows that Hanke-Pape-Schick's obstruction is dominated by a standard one, the Rosenberg index of M. We also extend our codimension 2 transfer map to secondary index invariants called the higher rho invariant. As a consequence, we obtain some example of psc manifolds are not psc null-cobordant.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/06/29

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Stability of non-proper functions (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Kenta Hayano**(Keio University)Stability of non-proper functions (JAPANESE)

[ Abstract ]

In this talk, we will give a sufficient condition for (strong) stability of non-proper functions (with respect to the Whitney topology). As an application, we will give a strongly stable but not infinitesimally stable function. We will further show that any Nash function on the Euclidean space becomes stable after a generic linear perturbation.

[ Reference URL ]In this talk, we will give a sufficient condition for (strong) stability of non-proper functions (with respect to the Whitney topology). As an application, we will give a strongly stable but not infinitesimally stable function. We will further show that any Nash function on the Euclidean space becomes stable after a generic linear perturbation.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/06/22

17:00-18:30 Online

Pre-registration required. See our seminar webpage.

On infinite presentations for the mapping class group of a compact non orientable surface and its twist subgroup (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Ryoma Kobayashi**(National Institute of Technology, Ishikawa College)On infinite presentations for the mapping class group of a compact non orientable surface and its twist subgroup (JAPANESE)

[ Abstract ]

An infinite presentation for the mapping class group of any compact orientable surface was given by Gervais, and then a simpler one by Luo. Using these results, an infinite presentation for the mapping class group of any compact non orientable surfaces with boundary less than or equal to one was given by Omori (Tokyo University of Science), and then one with boundary more than or equal to two by Omori and the speaker. In this talk, we first introduce an infinite presentation for the twisted subgroup of the mapping class group of any compact non orientable surface. I will also present four simple infinite presentations for the mapping group of any compact non orientable surface, which are an improvement of the one given by Omori and the speaker. This work includes a joint work with Omori.

[ Reference URL ]An infinite presentation for the mapping class group of any compact orientable surface was given by Gervais, and then a simpler one by Luo. Using these results, an infinite presentation for the mapping class group of any compact non orientable surfaces with boundary less than or equal to one was given by Omori (Tokyo University of Science), and then one with boundary more than or equal to two by Omori and the speaker. In this talk, we first introduce an infinite presentation for the twisted subgroup of the mapping class group of any compact non orientable surface. I will also present four simple infinite presentations for the mapping group of any compact non orientable surface, which are an improvement of the one given by Omori and the speaker. This work includes a joint work with Omori.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2021/06/15

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Direct decompositions of groups of piecewise linear homeomorphisms of the unit interval (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Takamichi Sato**(Waseda University)Direct decompositions of groups of piecewise linear homeomorphisms of the unit interval (JAPANESE)

[ Abstract ]

In this talk, we consider subgroups of the group PLo(I) of piecewise linear orientation-preserving homeomorphisms of the unit interval I = [0, 1] that are differentiable everywhere except at finitely many real numbers, under the operation of composition. We provide a criterion for any two subgroups of PLo(I) which are direct products of finitely many indecomposable non-commutative groups to be non-isomorphic. As its application we give a necessary and sufficient condition for any two subgroups of the R. Thompson group F that are stabilizers of finite sets of numbers in the interval (0, 1) to be isomorphic.

[ Reference URL ]In this talk, we consider subgroups of the group PLo(I) of piecewise linear orientation-preserving homeomorphisms of the unit interval I = [0, 1] that are differentiable everywhere except at finitely many real numbers, under the operation of composition. We provide a criterion for any two subgroups of PLo(I) which are direct products of finitely many indecomposable non-commutative groups to be non-isomorphic. As its application we give a necessary and sufficient condition for any two subgroups of the R. Thompson group F that are stabilizers of finite sets of numbers in the interval (0, 1) to be isomorphic.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html