Language：Others   Publishing type：Research paper (scientific journal)   Publisher：Journal of Topology and Analysis  

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.

DOI： 10.1142/S1793525321500680

Web of Science

Scopus

Language：Others   Publishing type：Research paper (scientific journal)   Publisher：Journal of Topology and Analysis  

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.

DOI： 10.1142/S1793525321500357

Web of Science

Scopus

Language：Others   Publishing type：Research paper (scientific journal)   Publisher：Bulletin of the London Mathematical Society  

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.

DOI： 10.1112/blms.12829

Web of Science

Scopus

Language：Others   Publishing type：Research paper (scientific journal)   Publisher：Israel Journal of Mathematics  

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.

DOI： 10.1007/s11856-022-2433-0

Web of Science

Scopus

Language：Others   Publishing type：Research paper (scientific journal)   Publisher：Homology, Homotopy and Applications  

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.

DOI： 10.4310/HHA.2023.V25.N1.A20

Web of Science

Scopus

Language：English   Publishing type：Research paper (scientific journal)  

DOI： https://doi.org/10.2206/kyushujm.74.197

Language：English   Publishing type：Research paper (scientific journal)  

A short proof for $mathrm{tc}(K)=4$
We show a method to determine topological complexity from the fibrewise view point, which provides an alternative proof for tc(K)=4, where K denotes Klein bottle.

DOI： 10.1016/j.topol.2019.06.014

Language：Others   Publishing type：Research paper (scientific journal)  

Homotopy pullback of A(n)-spaces and its applications to A(n)-types of gauge groups (vol 187, pg 1, 2015)
Introduction: The author regret that Section 9 of [5] contains a mistake, where we studied the classification problem of the gauge groups of principal [Formula presented]-bundles over [Formula presented]. In the proof of Proposition 9.1 in [5], the author considered the map [Formula presented] called the “relative Whitehead product”. But, actually, it is not well-defined. From this failure, the proofs for Proposition 9.1, Corollary 9.2 and Theorem 1.2 are no longer valid. The aim of this current article is to prove a weaker version of Theorem 1.2 in [5] and to improve the result for the fiberwise [Formula presented]-types of adjoint bundles. Let [Formula presented] be the principal [Formula presented]-bundle over [Formula presented] such that [Formula presented]. The following is a weaker version of Theorem 1.2 in [5], to which we only add the condition [Formula presented]. We denote the largest integer less than or equal to t by [Formula presented]. Theorem 1.1 For a positive integer [Formula presented], the gauge groups [Formula presented] and [Formula presented] are [Formula presented]-equivalent if [Formula presented] and [Formula presented] for any odd prime p. Moreover, if [Formula presented], the converse is also true. Proof To show the if part, it is sufficient to show that the wedge sum [Formula presented] extends over the product [Formula presented]. The case when [Formula presented] has already been verified in [4, Section 5]. Suppose [Formula presented]. By Toda's result [3, Section 7], we have homotopy groups of [Formula presented] as follows: [Formula presented] for [Formula presented], where [Formula presented] if [Formula presented]. This implies that, if [Formula presented] and [Formula presented], there is no obstruction to extending a map [Formula presented] over [Formula presented]. It also implies that, for [Formula presented] and a map [Formula presented], the composite [Formula presented] extends over [Formula presented]. Then we obtain the if part by induction and Theorem 1.1 in [5]. The proof of the converse in [5] correctly works for [Formula presented]. □ Remark 1.2 For [Formula presented] and [Formula presented], Toda's result [3, Theorem 7.5] says [Formula presented] This is the first non-trivial homotopy group where the obstruction is not detected in our method. Suppose there exists an extension [Formula presented] of [Formula presented], where [Formula presented] and i is the inclusion [Formula presented]. In the rest of this article, we compute the e-invariant [1] of the obstruction to extending the map f over [Formula presented]. This obstruction is regarded as an element [Formula presented]. The map h factors as the composite of the suspension map [Formula presented] and the inclusion [Formula presented], where [Formula presented] is the homotopy class corresponding to h under the isomorphism [Formula presented]. Consider the following maps among cofiber sequences: [Formula presented] As in [4], take the appropriate generator [Formula presented] such that [Formula presented] Actually, one can take the generator a as the image of [Formula presented] under the complexification map [Formula presented] from the quaternionic K-theory, where γ denotes the canonical line bundle and [Formula presented] the 1-dimensional trivial quaternionic vector bundle. We denote the restriction of a on [Formula presented] by [Formula presented]. Note that we can obtain the following by the Künneth theorem for K-theory: [Formula presented] Lemma 1.3 Let [Formula presented]. Then the following holds. (1) Suppose i is even. Then [Formula presented] is an image of the complexification map from the quaternionic K-theory.(2) Suppose i is odd. Then [Formula presented] is an image of the complexification map from the quaternionic K-theory, but [Formula presented] is not. Proof Consider the following commutative diagram induced by the cofiber sequence: [Formula presented] Note that all the groups appearing in this diagram are free abelian. This implies the vertical maps are injective. As is well-known, the index of the image of the map [Formula presented] is 1 if i is even, and is 2 if i is odd. Now the lemma follows from the above diagram and the fact that the image of [Formula presented] is generated by [Formula presented]. □ Since [Formula presented], there is a lift [Formula presented] of [Formula presented] contained in the image of the complexification from [Formula presented]. Denote the image of [Formula presented] under the map [Formula presented] by [Formula presented]. We take [Formula presented], [Formula presented] and [Formula presented]. We fix a generator of [Formula presented] such that its image in [Formula presented] is [Formula presented] We denote its images by [Formula presented] and [Formula presented]. As in [1, Section 7], the e-invariant λ of the map [Formula presented] is characterized by [Formula presented] in [Formula presented], where λ is well-defined as a residue class in [Formula presented] if n is odd, and in [Formula presented] if n is even. If the map [Formula presented] is null-homotopic, then λ is 0 as the corresponding residue class. By the result of [4], we have [Formula presented] where [Formula presented] and [Formula presented] are inductively defined by the equations [Formula presented] Since a is in the image of the complexification from [Formula presented], we have [Formula presented] for even [Formula presented] and [Formula presented] for odd [Formula presented] by Lemma 1.3. Combining with [4, Propositions 4.2 and 4.4], we have the following proposition. Lemma 1.4 The following hold. (1) For [Formula presented], [Formula presented].(2) For an odd prime p, [Formula presented]. There exists [Formula presented] such that the following holds: [Formula presented] Again by Lemma 1.3, [Formula presented] if n is odd. Note that the Chern characters ch a and [Formula presented] are computed as [Formula presented] Then, by computing [Formula presented] by two ways as in [4, Section 2], we obtain [Formula presented] By the definition of [Formula presented], we get [Formula presented] Then we have the following proposition from the e-invariant λ and Lemma 1.4. [Formula presented] Proposition 1.5 If f extends over [Formula presented], then the following hold. (1) For [Formula presented], [Formula presented].(2) For an odd prime p, [Formula presented]. Actually, nothing is improved by this proposition for odd p. But, for [Formula presented], we obtain the new result since the torsion part of [Formula presented] is annihilated by 4 [2, Corollary (1.22)]. Theorem 1.6 The adjoint bundle [Formula presented] is trivial as a fiberwise [Formula presented]-space if and only if k is divisible by [Formula presented]. From this result, one may expect that we can derive the classification of 2-local [Formula presented]-types of the gauge groups. But, to distinguish between [Formula presented] and [Formula presented] as [Formula presented]-spaces, we need some new technique. So, we leave this problem for now. The author would like to apologise for any inconvenience caused.

DOI： 10.1016/j.topol.2018.04.012

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.1017/S001708951600063X

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.1016/j.topol.2017.05.012

Language：Others   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.4310/HHA.2019.v21.n1.a6

Language：English   Publishing type：Research paper (scientific journal)  

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 < infinity and K is a finite complex. We show that the number of A(infinity)-types of the gauge groups of P is infinite if K is a sphere and there are infinitely many P.

DOI： 10.1112/jtopol/jtv025

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.1016/j.topol.2015.02.014

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.4153/CMB-2013-038-8

Language：Others  

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 $p$.

DOI： 10.1017/S001708951600063X

Language：English   Publishing type：Research paper (scientific journal)  

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 <= d <= p - 1 and n <= 2p - 1, we give a concrete classification of their p-local homotopy types.

DOI： 10.1017/S0308210512001278

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.1016/j.jpaa.2011.10.020

Language：English   Publishing type：Research paper (scientific journal)  

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 < infinity. As an example, we give a lower bound of the number of A(n)-equivalence types of gauge groups of principal SU(2)-bundles over S-4.

DOI： 10.1112/jlms/jdr040

Event date： 2019.6

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：京都大学理学研究科数学教室   Country：Japan  

Event date： 2018.12

Language：Japanese   Presentation type：Oral presentation (general)  

Country：Japan  

Event date： 2018.9

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：沖縄県青年会館   Country：Japan  

Event date： 2018.8

Language：English   Presentation type：Oral presentation (general)  

Venue：Kyoto University   Country：Japan  

An-maps are morphisms between An-spaces introduced by Sugawara, Stasheff, Boardman-Vogt and Iwase. Sugawara, Stasheff and Iwase characterised the condition when a map between An-spaces admits an An-map structure in terms of projective spaces. In this talk, we see that a refinement of this result is realised as a weak homotopy equivalence between the space of An-maps An(G,H) and the space of based maps Map∗(BnG,BH) from the n-th projective space BnG to the classifying space BH. We also see some applications of this results to extension of an evaluation fibration and homotopy commutativity.

Event date： 2018.6

Language：English   Presentation type：Oral presentation (general)  

Venue：Sabhal Mòr Ostaig   Country：United Kingdom  

Event date： 2018.1

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：福岡大学   Country：Japan  

Event date： 2017.11

Language：Japanese   Presentation type：Oral presentation (general)  

Country：Japan  

Event date： 2017.8

Language：English   Presentation type：Oral presentation (general)  

Venue：Kyoto University   Country：Japan  

Event date： 2017.3

Language：Japanese   Presentation type：Oral presentation (invited, special)  

Venue：首都大学東京   Country：Japan  

Event date： 2017.2

Language：English   Presentation type：Oral presentation (general)  

Venue：Autonomous University of Barcelona   Country：Spain  

Event date： 2017.2

Language：English   Presentation type：Oral presentation (general)  

Venue：University of Malaga   Country：Spain  

Event date： 2016.12

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：京都大学   Country：Japan  

Event date： 2016.12

Language：Japanese   Presentation type：Oral presentation (general)  

Country：Japan  

Event date： 2016.11

Language：Japanese   Presentation type：Oral presentation (general)  

Country：Japan  

Event date： 2016.9

Language：Japanese   Presentation type：Oral presentation (general)  

Country：Japan  

Event date： 2016.9

Language：Japanese   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

Venue：京都大学   Country：Japan  

Event date： 2016.7

Language：Japanese   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

Venue：愛媛大学   Country：Japan  

Event date： 2016.5 - 2016.7

Language：Japanese   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

Venue：福岡大学   Country：Japan  

Language：Japanese   Presentation type：Oral presentation (general)  

Country：Japan  

The A_n-structure (n=1,2,...,∞) of a topological group describes certain higher homotopy structure concerned with its binary operation. It has some relation with a generalization of projective spaces and homotopy invariants such as LS-category. The notion

Event date： 2023.6

Language：Japanese   Presentation type：Oral presentation (general)  

Event date： 2023.2

Language：English   Presentation type：Oral presentation (general)  

Venue：大阪公立大学   Country：Japan  

An associative model of homotopy coherent functors and natural transforamations

Event date： 2022.9

Language：English   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

Venue：University of Aberdeen   Country：United Kingdom  

Event date： 2022.9

Language：English   Presentation type：Oral presentation (general)  

Venue：ICMS, Bayes Centre, the University of Edinburgh   Country：United Kingdom  

Event date： 2021.12

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：オンライン   Country：Japan  

正規部分群はcrossed moduleと呼ばれる対象に一般化される．crossed moduleは準同型 H -> G とそれと整合的な「共役作用」 G -> Aut(H) からなるようなものである．正規部分群による商が群になることの一般化として，Farjoun-Segevはtopological crossed moduleのホモトピー商が自然に位相群になることを示した．本講演では高次ホモトピーを用いたcrossed moduleの一般化であるN_k(l)-map (k,l ≥ 1)を導入する．これはMcCartyやJamesにより定義された古典的なホモトピー正規性よりも強い性質となっている．N_k(l)-mapはファイバーワイズ射影空間を用いた特徴付けを持つ．また，N_k(l)-mapはそのホモトピー商のLSカテゴリーが小さいとき，H-空間となることを示す．さらに応用として，包含写像 SU(m) -> SU(n) がいつp-局所的にN_k(l)-mapとなるのか調べる．

Event date： 2021.6

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：東京都立大学（オンライン）   Country：Japan  

Language：Japanese  

Venue：奈良女子大学   Country：Japan  

正規部分群による商が群になることの一般化として, Farjoun-Segev は topological crossed module のホモトピー商が自然に位相群になることを示した. 本講演では高次ホモトピーを用いた crossed module の一般化である Nk(l)-map (k, l ≥ 1) を導入する. これは McCarty や James により定義された古典的なホモトピー正規性よりも強い性質となっている. Nk(l)-map はファイバーワイズ射影空間を用いたよい特徴付けを持つ. また, Nk(l)-map のホモトピー商は LS カテゴリーが小さいとき, H-空間となることを示す. さらに応用として, 包含写像 SU(m) → SU(n) がいつ p-局所的に Nk(l)-map となるのか調べる.

Language：English  

Venue：オンライン   Country：Japan  

Language：Japanese  

Country：Japan  

Language：English  

Venue：HUS, Vietnam National University  

Language：Japanese   Presentation type：Oral presentation (general)  

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：大阪公立大学  

Language：English   Presentation type：Oral presentation (general)  

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：奈良女子大学  

Language：Others  

Given an action of a finite group &#36;G&#36;, we can define its index. The &#36;G&#36;-index
roughly measures a size of the given &#36;G&#36;-space. We explore connections between
the &#36;G&#36;-index theory and topological dynamics. For a fixed-point free dynamical
system, we study the &#36;mathbb{Z}_p&#36;-index of the set of &#36;p&#36;-periodic points. We
find that its growth is at most linear in &#36;p&#36;. 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.

Language：Others  

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 &#36;pi_0&#36; 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
&#36;mathbb{C}&#36;-valued &#36;mathbb{Z}&#36;-sequences, so we can determine its homotopy
groups. We also study the space of (end-)periodic finite propagation unitary
operators.

Language：Others  

The first aim of this paper is to study the &#36;p&#36;-local higher homotopy
commutativity of Lie groups in the sense of Sugawara. The second aim is to
apply this result to the &#36;p&#36;-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
&#36;5&#36;-local higher homotopy non-commutativity problem of the exceptional Lie
group &#36;mathrm{G}_2&#36;, which has been open for a long time.

DOI： 10.4310/HHA.2019.v21.n1.a6

Language：Others  

We give a homotopy invariant construction of the Reidemeister trace for the
coincidence of two maps between closed manifolds of not necessarily the same
dimensions. It is realized as a homology class of the homotopy equalizer, which
coincides with the Hurewicz image of Koschorke's stabilized bordism invariant.
To define it, we use a kind of shriek maps appearing string topology. As an
application, we compute the coincidence Reidemeister trace for the
self-coincidence of the projections of &#36;S^1&#36;-bundles on &#36;mathbb{C}P^n&#36;. We
also mention how to relate our construction to the string topology operation
called the loop coproduct.

Language：English  

ON EVALUATION FIBER SEQUENCES (The Topology and the Algebraic Structures of Transformation Groups)

Language：English  

ON EVALUATION FIBER SEQUENCES (The Topology and the Algebraic Structures of Transformation Groups)

Language：Others  

A Lie group is called &#36;p&#36;-regular if it has the &#36;p&#36;-local homotopy type of a
product of spheres. (Non)triviality of the Samelson products of the inclusions
of the factor spheres into &#36;p&#36;-regular &#36;mathrm{SO}(2n)_{(p)}&#36; is determined,
which completes the list of (non)triviality of such Samelson products in
&#36;p&#36;-regular simple Lie groups. As an application, we determine the homotopy
normality of the inclusion &#36;mathrm{SO}(2n-1) omathrm{SO}(2n)&#36; in the sense
of James at any prime &#36;p&#36;.

DOI： 10.1017/S001708951600063X

Language：English  

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 <= d <= p - 1 and n <= 2p - 1, we give a concrete classification of their p-local homotopy types.

DOI： 10.1017/S0308210512001278