Tsuyoshi Kato
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Tsuyoshi Kato has a total of 13 co-authors. Here are the top co-authors by number of shared publications:
- Daisuke Kishimoto
Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan
8 shared publications - Mitsunobu Tsutaya
Faculty of Mathematics, Kyushu University, Fukuoka, Japan
6 shared publications - Satoshi Tsujimoto
Graduate School of Informatics, Kyoto University, Kyoto, Japan
4 shared publications - Kouichi Yasui3 shared publications
- Nobuhiro Nakamura
Integrated Center for Science and Humanities, Fukushima Medical University, Fukushima, Japan
3 shared publications - Andrzej Zuk
Institut de Mathematiques, Paris, France
2 shared publications
Latest Publications
- Open Access
Notions of simple type for Bauer--Furuta invariants
By extending the notion of simple type for the Seiberg--Witten invariant of a 4-manifold, we introduce notions of BF blowup simple type and BF homogeneous type for the Bauer--Furuta invariant and study their applications. Specifically, we show that the existence of an immersed 2-sphere with a certain condition guarantees BF blowup simple type. As an application, we determine the Bauer--Furuta invariant of a 4-manifold obtained by a logarithmic transformation along a torus in a fishtail neighborhood. We also give constraints on gluing decompositions of 4-manifolds by using BF homogeneous type. To prove these results, we also give gluing formulae and an immersed adjunction inequality for Bauer--Furuta invariants.
Open AccessNotions of simple type for Bauer--Furuta invariants
By extending the notion of simple type for the Seiberg--Witten invariant of a 4-manifold, we introduce notions of BF blowup simple type and BF homogeneous type for the Bauer--Furuta invariant and study their applications. Specifically, we show that the existence of an immersed 2-sphere with a certain condition guarantees BF blowup simple type. As an application, we determine the Bauer--Furuta invariant of a 4-manifold obtained by a logarithmic transformation along a torus in a fishtail neighborhood. We also give constraints on gluing decompositions of 4-manifolds by using BF homogeneous type. To prove these results, we also give gluing formulae and an immersed adjunction inequality for Bauer--Furuta invariants.
- Open Access
Uniform Lefschetz fixed-point theory
We develop the Lefschetz fixed-point theory for noncompact manifolds of bounded geometry and uniformly continuous maps. Specifically, we define the uniform Lefschetz class $\mathscr{L}(f)$ of a uniformly continuous map $f\colon M\to M$ of a uniform simply-connected noncompact complete Riemannian manifold of bounded geometry $M$ satisfying $d(f,1)<\infty$, and prove that $\mathscr{L}(f)=0$ if and only if $f$ is uniformly homotopic to a strongly fixed-point free (without fixed-points on $M$ and at infinity) uniformly continuous map. To achieve this, we introduce a new cohomology for metric spaces, called uniform bounded cohomology, which is a variant of bounded cohomology, and develop an obstruction theory formulated in terms of this cohomology.Open AccessUniform Lefschetz fixed-point theory
We develop the Lefschetz fixed-point theory for noncompact manifolds of bounded geometry and uniformly continuous maps. Specifically, we define the uniform Lefschetz class $\mathscr{L}(f)$ of a uniformly continuous map $f\colon M\to M$ of a uniform simply-connected noncompact complete Riemannian manifold of bounded geometry $M$ satisfying $d(f,1)<\infty$, and prove that $\mathscr{L}(f)=0$ if and only if $f$ is uniformly homotopic to a strongly fixed-point free (without fixed-points on $M$ and at infinity) uniformly continuous map. To achieve this, we introduce a new cohomology for metric spaces, called uniform bounded cohomology, which is a variant of bounded cohomology, and develop an obstruction theory formulated in terms of this cohomology. - Open Access
- 9 December 2024
- Published by Mathematical Sciences Publishers in Algebraic & Geometric Topology
- Vol. 24 (7) , 3985-3996
Vector fields on noncompact manifolds
Let M be a noncompact connected manifold with a cocompact and properly discontinuous action of a discrete group G. We establish a Poincaré-Hopf theorem for a bounded vector field on M satisfying a mild condition on zeros. As an application, we show that such a vector field must have infinitely many zeros whenever G is amenable and the Euler characteristic of M=G is nonzero. 57R25, 58K45
- 9 December 2024
- Published by Mathematical Sciences Publishers in Algebraic & Geometric Topology
- Vol. 24 (7) , 3985-3996
Open AccessVector fields on noncompact manifolds
Let M be a noncompact connected manifold with a cocompact and properly discontinuous action of a discrete group G. We establish a Poincaré-Hopf theorem for a bounded vector field on M satisfying a mild condition on zeros. As an application, we show that such a vector field must have infinitely many zeros whenever G is amenable and the Euler characteristic of M=G is nonzero. 57R25, 58K45
- Open Access
Vector fields on non-compact manifolds
Let $M$ be a non-compact connected manifold with a cocompact and properly discontinuous action of a discrete group $G$. We establish a Poincar\'{e}-Hopf theorem for a bounded vector field on $M$ satisfying a mild condition on zeros. As an application, we show that such a vector field must have infinitely many zeros whenever $G$ is amenable and the Euler characteristic of $M/G$ is non-zero.Open AccessVector fields on non-compact manifolds
Let $M$ be a non-compact connected manifold with a cocompact and properly discontinuous action of a discrete group $G$. We establish a Poincar\'{e}-Hopf theorem for a bounded vector field on $M$ satisfying a mild condition on zeros. As an application, we show that such a vector field must have infinitely many zeros whenever $G$ is amenable and the Euler characteristic of $M/G$ is non-zero. - Open AccessHomotopy type of the space of finite propagation unitary operators on $\mathbb{Z}$The index theory for the space of finite propagation unitary operators was developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks in mathematical physics. In particular, they proved that $\pi_0$ of the space is determined by the index. However, nothing is known about the higher homotopy groups. In this article, we describe the homotopy type of the space of finite propagation unitary operators on the Hilbert space of square summable $\mathbb{C}$-valued $\mathbb{Z}$-sequences, so we can determine its homotopy groups. We also study the space of (end-)periodic finite propagation unitary operators.Open AccessHomotopy type of the space of finite propagation unitary operators on $\mathbb{Z}$The index theory for the space of finite propagation unitary operators was developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks in mathematical physics. In particular, they proved that $\pi_0$ of the space is determined by the index. However, nothing is known about the higher homotopy groups. In this article, we describe the homotopy type of the space of finite propagation unitary operators on the Hilbert space of square summable $\mathbb{C}$-valued $\mathbb{Z}$-sequences, so we can determine its homotopy groups. We also study the space of (end-)periodic finite propagation unitary operators.
- 14 March 2023
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 151 (06) , 2695-2705
A note on exotic families of 4-manifolds
We present a pair of smooth fiber bundles over the circle with a common 4 4 -dimensional fiber with the following properties: (1) their total spaces are diffeomorphic to each other; (2) they are isomorphic to each other as topological fiber bundles; (3) they are not isomorphic to each other as smooth fiber bundles. In particular, we exhibit an example with non-simply-connected fiber.
- 14 March 2023
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 151 (06) , 2695-2705
A note on exotic families of 4-manifolds
We present a pair of smooth fiber bundles over the circle with a common 4 4 -dimensional fiber with the following properties: (1) their total spaces are diffeomorphic to each other; (2) they are isomorphic to each other as topological fiber bundles; (3) they are not isomorphic to each other as smooth fiber bundles. In particular, we exhibit an example with non-simply-connected fiber.
- Open Access
Upper bounds for virtual dimensions of Seiberg-Witten moduli spaces
Given a closed four-manifold with $b_1=0$ and a prime number $p$, we prove that for any mod $p^r$ basic class, the virtual dimension of the Seiberg-Witten moduli space is bounded above by $2r(p-1)-2$ under some conditions on $r$ and $b_2^+$. As an application, we obtain adjunction inequalities for embedded surfaces with negative self-intersection number.Open AccessUpper bounds for virtual dimensions of Seiberg-Witten moduli spaces
Given a closed four-manifold with $b_1=0$ and a prime number $p$, we prove that for any mod $p^r$ basic class, the virtual dimension of the Seiberg-Witten moduli space is bounded above by $2r(p-1)-2$ under some conditions on $r$ and $b_2^+$. As an application, we obtain adjunction inequalities for embedded surfaces with negative self-intersection number. - Open Access
- 1 January 2023
- Published by International Press of Boston in Homology, Homotopy and Applications
- Vol. 25 (1) , 375-400
Homotopy type of the space of finite propagation unitary operators on $\mathbb{Z}$International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- 1 January 2023
- Published by International Press of Boston in Homology, Homotopy and Applications
- Vol. 25 (1) , 375-400
Open AccessHomotopy type of the space of finite propagation unitary operators on $\mathbb{Z}$International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- Open Access
$L^2$ harmonic theory, Seiberg-Witten theory and asymptotics of differential forms
We present a pair of open smooth $4$-manifolds that are mutually homeomorphic. One of them admits a Riemannian metric that possesses quasi-cylindricity, and positivity of scalar curvature and of dimension of certain $L^2$ harmonic forms. By contrast, for the other manifold, no Riemannian metric can simultaneously satisfy these properties. Our method uses Seiberg-Witten theory on compact $4$-manifolds and applies $L^2$ harmonic theory on non-compact, complete Riemannian $4$-manifolds. We introduce a new argument to apply Gauge theory, which arises from a discovery of an asymptotic property of the range of the differential.Open Access$L^2$ harmonic theory, Seiberg-Witten theory and asymptotics of differential forms
We present a pair of open smooth $4$-manifolds that are mutually homeomorphic. One of them admits a Riemannian metric that possesses quasi-cylindricity, and positivity of scalar curvature and of dimension of certain $L^2$ harmonic forms. By contrast, for the other manifold, no Riemannian metric can simultaneously satisfy these properties. Our method uses Seiberg-Witten theory on compact $4$-manifolds and applies $L^2$ harmonic theory on non-compact, complete Riemannian $4$-manifolds. We introduce a new argument to apply Gauge theory, which arises from a discovery of an asymptotic property of the range of the differential. - Open Access
Hilbert bundles with ends
Given a countable metric space, we can consider its end. Then a basis of a Hilbert space indexed by the metric space defines an end of the Hilbert space, which is a new notion and different from an end as a metric space. Such an indexed basis also defines unitary operators of finite propagation, and these operators preserve an end of a Hilbert space. Then, we can define a Hilbert bundle with end, which lightens up new structures of Hilbert bundles. In a special case, we can define characteristic classes of Hilbert bundles with ends, which are new invariants of Hilbert bundles. We show Hilbert bundles with ends appear in natural contexts. First, we generalize the pushforward of a vector bundle along a finite covering to an infinite covering, which is a Hilbert bundle with end under a mild condition. Then we compute characteristic classes of some pushforwards along infinite coverings. Next, we will show the spectral decompositions of nice differential operators give rise to Hilbert bundles with ends, which elucidate new features of spectral decompositions. The spectral decompositions we will consider are the Fourier transform and the harmonic oscillators.
Open AccessHilbert bundles with ends
Given a countable metric space, we can consider its end. Then a basis of a Hilbert space indexed by the metric space defines an end of the Hilbert space, which is a new notion and different from an end as a metric space. Such an indexed basis also defines unitary operators of finite propagation, and these operators preserve an end of a Hilbert space. Then, we can define a Hilbert bundle with end, which lightens up new structures of Hilbert bundles. In a special case, we can define characteristic classes of Hilbert bundles with ends, which are new invariants of Hilbert bundles. We show Hilbert bundles with ends appear in natural contexts. First, we generalize the pushforward of a vector bundle along a finite covering to an infinite covering, which is a Hilbert bundle with end under a mild condition. Then we compute characteristic classes of some pushforwards along infinite coverings. Next, we will show the spectral decompositions of nice differential operators give rise to Hilbert bundles with ends, which elucidate new features of spectral decompositions. The spectral decompositions we will consider are the Fourier transform and the harmonic oscillators.