| 1. |
Mitsunobu Tsutaya, Higher homotopy normalities in topological groups, Journal of Topology, 10.1112/topo.12282, 16, 1, 234-263, 2021.11, The purpose of this paper is to introduce $N_k(ell)$-maps ($1le
k,ellleinfty$), which describe higher homotopy normalities, and to study
their basic properties and examples. $N_k(ell)$-map is defined with higher
homotopical conditions. It is shown that a homomorphism is an $N_k(ell)$-map
if and only if there exists fiberwise maps between fiberwise projective spaces
with some properties. Also, the homotopy quotient of an $N_k(k)$-map is shown
to be an $H$-space if its LS category is not greater than $k$. As an
application, we investigate when the inclusions
$operatorname{SU}(m) ooperatorname{SU}(n)$ and
$operatorname{SO}(2m+1) ooperatorname{SO}(2n+1)$ are $p$-locally
$N_k(ell)$-maps..  |
| 2. |
Tsuyoshi Kato, Daisuke Kishimoto, Mitsunobu Tsutaya, Hilbert bundles with ends, Journal of Topology and Analysis, 10.1142/S1793525321500680, 2021.05, 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.. |
| 3. |
Tsuyoshi Kato, Daisuke Kishimoto, Mitsunobu Tsutaya, Homotopy type of the unitary group of the uniform Roe algebra on $mathbb{Z}^n$, Journal of Topology and Analysis, 10.1142/S1793525321500357, 15, 2, 495-512, 2021.02, We study the homotopy type of the space of the unitary group
$operatorname{U}_1(C^ast_u(|mathbb{Z}^n|))$ of the uniform Roe algebra
$C^ast_u(|mathbb{Z}^n|)$ of $mathbb{Z}^n$. We show that the stabilizing map
$operatorname{U}_1(C^ast_u(|mathbb{Z}^n|)) ooperatorname{U}_infty(C^ast_u(|mathbb{Z}^n|))$
is a homotopy equivalence. Moreover, when $n=1,2$, we determine the homotopy
type of $operatorname{U}_1(C^ast_u(|mathbb{Z}^n|))$, which is the product of
the unitary group $operatorname{U}_1(C^ast(|mathbb{Z}^n|))$ (having the
homotopy type of $operatorname{U}_infty(mathbb{C})$ or $mathbb{Z} imes
Boperatorname{U}_infty(mathbb{C})$ depending on the parity of $n$) of the
Roe algebra $C^ast(|mathbb{Z}^n|)$ and rational Eilenberg--MacLane spaces.. |
| 4. |
Sho Hasui, Daisuke Kishimoto, Masahiro Takeda, Mitsunobu Tsutaya, Tverberg's theorem for cell complexes, Bulletin of the London Mathematical Society, 10.1112/blms.12829, 55, 4, 1944-1956, 2021.01, The topological Tverberg theorem states that any continuous map of a
$(d+1)(r-1)$-simplex into the Euclidean $d$-space maps some points from $r$
pairwise disjoint faces of the simplex to the same point whenever $r$ is a
prime power. We substantially generalize this theorem to continuous maps of
certain CW complexes, including simplicial $((d+1)(r-1)-1)$-spheres, into the
Euclidean $d$-space. We also discuss the atomicity of the Tverberg property.. |
| 5. |
Masaki Tsukamoto, Mitsunobu Tsutaya, Masahiko Yoshinaga, $G$-index, topological dynamics and marker property, Israel Journal of Mathematics, 10.1007/s11856-022-2433-0, 251, 2, 737-764, 2020.12, Given an action of a finite group $G$, we can define its index. The $G$-index
roughly measures a size of the given $G$-space. We explore connections between
the $G$-index theory and topological dynamics. For a fixed-point free dynamical
system, we study the $mathbb{Z}_p$-index of the set of $p$-periodic points. We
find that its growth is at most linear in $p$. As an application, we construct
a free dynamical system which does not have the marker property. This solves a
problem which has been open for several years.. |
| 6. |
Tsuyoshi Kato, Daisuke Kishimoto, Mitsunobu Tsutaya, Homotopy type of the space of finite propagation unitary operators on $mathbb{Z}$, Homology, Homotopy and Applications, 10.4310/HHA.2023.V25.N1.A20, 25, 1, 375-400, 2020.07, 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.. |
| 7. |
Norio Iwase, Mitsunobu Tsutaya, UPPER BOUND FOR MONOIDAL TOPOLOGICAL COMPLEXITY, Kyushu Journal of Mathematics, https://doi.org/10.2206/kyushujm.74.197, 74, 1, 197-200, 2020.01. |
| 8. |
Iwase Norio, Sakai Michihiro, Tsutaya Mitsunobu, A short proof for $mathrm{tc}(K)=4$, Topology and its Applications, 10.1016/j.topol.2019.06.014, 264, 167-174, 2019.09. |
| 9. |
Tsutaya Mitsunobu, Homotopy pullback of A(n)-spaces and its applications to A(n)-types of gauge groups (vol 187, pg 1, 2015), TOPOLOGY AND ITS APPLICATIONS, 10.1016/j.topol.2018.04.012, 243, 159-162, 2018.07. |
| 10. |
Daisuke Kishimoto, Mitsunobu Tsutaya, SAMELSON PRODUCTS IN p-REGULAR SO(2n) ANDITS HOMOTOPY NORMALITY, GLASGOW MATHEMATICAL JOURNAL, 10.1017/S001708951600063X, 60, 1, 165-174, 2018.01, A Lie group is called p-regular if it has the p-local homotopy type of a product of spheres. (Non) triviality of the Samelson products of the inclusions of the factor spheres into p-regular SO(2n)((p)) is determined, which completes the list of (non) triviality of such Samelson products in p-regular simple Lie groups. As an application, we determine the homotopy normality of the inclusion SO(2n - 1) -> SO(2n) in the sense of James at any prime p.. |
| 11. |
Daisuke Kishimoto, Stephen Theriault, Mitsunobu Tsutaya, The homotopy types of G(2)-gauge groups, TOPOLOGY AND ITS APPLICATIONS, 10.1016/j.topol.2017.05.012, 228, 92-107, 2017.09, The equivalence class of a principal G(2)-bundle over S-4 is classified by the value k is an element of Z of the second Chern class. In this paper we consider the homotopy types of the corresponding gauge groups g(k), and determine the number of homotopy types up to one factor of 2. (C) 2017 Elsevier B.V. All rights reserved.. |
| 12. |
Sho Hasui, Daisuke Kishimoto, Mitsunobu Tsutaya, Higher homotopy commutativity in localized Lie groups and gauge groups, Homology, Homotopy and Applications, 10.4310/HHA.2019.v21.n1.a6, 21, 1, 107-128, 2016.12, The first aim of this paper is to study the $p$-local higher homotopy
commutativity of Lie groups in the sense of Sugawara. The second aim is to
apply this result to the $p$-local higher homotopy commutativity of gauge
groups. Although the higher homotopy commutativity of Lie groups in the sense
of Williams is already known, the higher homotopy commutativity in the sense of
Sugawara is necessary for this application. The third aim is to resolve the
$5$-local higher homotopy non-commutativity problem of the exceptional Lie
group $mathrm{G}_2$, which has been open for a long time.. |
| 13. |
Daisuke Kishimoto, Mitsunobu Tsutaya, Infiniteness of A(infinity)-types of gauge groups, JOURNAL OF TOPOLOGY, 10.1112/jtopol/jtv025, 9, 1, 181-191, 2016.03, Let G be a compact connected Lie group and let P be a principal G-bundle over K. The gauge group of P is the topological group of automorphisms of P. For fixed G and K, consider all principal G-bundles P over K. It is proved in Crabb and Sutherland [ Proc. London Math. Soc. (3) 81 (2000) 747-768] and Tsutaya [ J. London Math. Society 85 (2012) 142-164] that the number of An-types of the gauge groups of P is finite if n |
| 14. |
Mitsunobu Tsutaya, Homotopy pullback of A(n)-spaces and its applications to A(n)-types of gauge groups, TOPOLOGY AND ITS APPLICATIONS, 10.1016/j.topol.2015.02.014, 187, 1-25, 2015.06, We construct the homotopy pullback of A(n)-spaces and show some universal property of it. As the first application, we review Zabrodsky's result which states that for each prime p, there is a finite CW complex which admits an A(p-1)-form but no A(p)-form. As the second application, we investigate A(n)-types of gauge groups. In particular, we give a new result on A(n)-types of the gauge groups of principal SU(2)-bundles over S-4, which is a complete classification when they are localized away from 2. (C) 2015 Elsevier B.V. All rights reserved.. |
| 15. |
Daisuke Kishimoto, Akira Kono, Mitsunobu Tsutaya, On Localized Unstable K-1-groups and Applications to Self-homotopy Groups, CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 10.4153/CMB-2013-038-8, 57, 2, 344-356, 2014.06, The method for computing the p-localization of the group [X, U(n)], by Hamanaka in 2004, is revised. As an application, an explicit description of the self-homotopy group of Sp(3) localized at p >= 5 is given and the homotopy nilpotency of Sp(3) localized at p >= 5 is determined.. |
| 16. |
Daisuke Kishimoto, Mitsunobu Tsutaya, Samelson products in p-regular SO(2n) and its homotopy normality, 10.1017/S001708951600063X, 2014.05, A Lie group is called $p$-regular if it has the $p$-local homotopy type of a
product of spheres. (Non)triviality of the Samelson products of the inclusions
of the factor spheres into $p$-regular $mathrm{SO}(2n)_{(p)}$ is determined,
which completes the list of (non)triviality of such Samelson products in
$p$-regular simple Lie groups. As an application, we determine the homotopy
normality of the inclusion $mathrm{SO}(2n-1) omathrm{SO}(2n)$ in the sense
of James at any prime $p$.. |
| 17. |
Daisuke Kishimoto, Akira Kono, Mitsunobu Tsutaya, On p-local homotopy types of gauge groups, PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, 10.1017/S0308210512001278, 144, 1, 149-160, 2014.02, The aim of this paper is to show that the p-local homotopy type of the gauge group of a principal bundle over an even-dimensional sphere is completely determined by the divisibility of the classifying map by p. In particular, for gauge groups of principal SU(n)-bundles over S-2d for 2 |
| 18. |
Mitsunobu Tsutaya, A note on homotopy types of connected components of Map (S-4, BSU (2)), JOURNAL OF PURE AND APPLIED ALGEBRA, 10.1016/j.jpaa.2011.10.020, 216, 4, 826-832, 2012.04, Gottlieb has shown that connected components of Map (S-4, BSU(2)) are the classifying spaces of gauge groups of principal SU(2)-bundles over S-4. Tsukuda has investigated the homotopy types of connected components of Map (S-4, BSU(2)). But unfortunately, his proof is not complete for p = 2. In this paper, we give a complete proof. Moreover, we investigate the further divisibility of epsilon(i) defined in Tsukuda's paper. We apply this to classification problem of gauge groups as A(n)-spaces. (C) 2011 Elsevier B.V. All rights reserved.. |
| 19. |
Mitsunobu Tsutaya, Finiteness of A(n)-equivalence types of gauge groups, JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 10.1112/jlms/jdr040, 85, 1, 142-164, 2012.02, Let B be a finite CW complex and G be a compact connected Lie group. We show that the number of gauge groups of principal G-bundles over B is finite up to A(n)-equivalence for n |
