TY - JOUR
T1 - Corrigendum to ‘Homotopy pullback of An-spaces and its applications to An-types of gauge groups’ [Topology Appl. 187 (2015) 1–25] (S0166864115000838) (10.1016/j.topol.2015.02.014))
AU - Tsutaya, Mitsunobu
PY - 2018/7/1
Y1 - 2018/7/1
N2 - 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 From this failure, the proofs for Proposition 9.1, Corollary 9.2 and Theorem 1.2 are no longer valid.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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85047544925&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85047544925&partnerID=8YFLogxK
U2 - 10.1016/j.topol.2018.04.012
DO - 10.1016/j.topol.2018.04.012
M3 - Comment/debate
AN - SCOPUS:85047544925
VL - 243
SP - 159
EP - 162
JO - Topology and its Applications
JF - Topology and its Applications
SN - 0166-8641
ER -