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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. |
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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. |
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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 From this failure, the proofs for Proposition 9.1, Corollary 9.2 and Theorem 1.2 are no longer valid.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.. |
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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.. |
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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.. |
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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 |
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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.. |
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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.. |
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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 |
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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.. |
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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 |
