Homotopy nilpotency in p-regular loop spaces

Kaji, Shizuo; Kishimoto, Daisuke

doi:10.1007/s00209-008-0459-6

Cited by 27 publications

(22 citation statements)

References 29 publications

(56 reference statements)

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“…One easily sees that in investigating the multiplicative structure of G (p) , the Samelson products i , j play the fundamental role as in [9], where i is the inclusion…”

Section: A ∧ B → X (X Y) → α(X)β(y)α(x) −1 β(Y)mentioning

confidence: 99%

“…Some of these Samelson products are calculated in [5,9], and (non)triviality of all these Samelson products is determined in [6] as follows. Thus, the only remaining case is SO(2n).…”

Section: A ∧ B → X (X Y) → α(X)β(y)α(x) −1 β(Y)mentioning

confidence: 99%

“…By Lemma 2.3, if P 1 x k includes the term cx i x j with c = 0, then the Samelson product i , j in G (p) is non-trivial. As in [9], if i , j is non-trivial, then for some 1 ≤ k ≤ we have n k + p − 1 = n i + n j and π k • i , j is non-trivial. Again by Lemma 2.3, this implies that P 1 x k includes the term cx i x j with c = 0.…”

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confidence: 90%

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SAMELSON PRODUCTS IN p-REGULAR SO(2n) AND ITS HOMOTOPY NORMALITY

Kishimoto

Tsutaya²

2017

Glasgow Math. J.

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Abstract.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.2010 Mathematics Subject Classification. 55Q15.

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“…One easily sees that in investigating the multiplicative structure of G (p) , the Samelson products i , j play the fundamental role as in [9], where i is the inclusion…”

Section: A ∧ B → X (X Y) → α(X)β(y)α(x) −1 β(Y)mentioning

confidence: 99%

“…Some of these Samelson products are calculated in [5,9], and (non)triviality of all these Samelson products is determined in [6] as follows. Thus, the only remaining case is SO(2n).…”

Section: A ∧ B → X (X Y) → α(X)β(y)α(x) −1 β(Y)mentioning

confidence: 99%

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SAMELSON PRODUCTS IN p-REGULAR SO(2n) AND ITS HOMOTOPY NORMALITY

Kishimoto

Tsutaya²

2017

Glasgow Math. J.

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“…In [5], Kaji and the author approached this question by considering homotopy nilpotency which is defined as follows, where we treat only group-like spaces (see [15] for a general definition). Let X be a group-like space, that is, X satisfies all the axioms of groups up to homotopy, and let γ : X × X → X be the commutator map of X.…”

Section: Question 12 How Far From Being Homotopy Commutative Is G (mentioning

confidence: 99%

“…In [5], Kaji and the author determined nil X for a p-compact group X when p is a regular prime, that is, X has the homotopy type of the direct product of localized spheres. For example, they showed nil SU(n) (p) = 2 for 3 2 n < p < 2n 3 for n p when p is odd, and nil SU(2) (2) = 2.…”

Section: Question 12 How Far From Being Homotopy Commutative Is G (mentioning

confidence: 99%

Homotopy nilpotency in localized $SU(n)$

Kishimoto

2009

Homology, Homotopy and Applications

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Homotopy nilpotency of some homogeneous spaces

Golasiński

2021

manuscripta math.

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Let $${\mathbb {K}}={\mathbb {R}},\,{\mathbb {C}}$$ K = R , C , the field of reals or complex numbers and $${\mathbb {H}}$$ H , the skew $${\mathbb {R}}$$ R -algebra of quaternions. We study the homotopy nilpotency of the loop spaces $$\Omega (G_{n,m}({\mathbb {K}}))$$ Ω ( G n , m ( K ) ) , $$\Omega (F_{n;n_1,\ldots ,n_k}({\mathbb {K}}))$$ Ω ( F n ; n 1 , … , n k ( K ) ) , and $$\Omega (V_{n,m}({\mathbb {K}}))$$ Ω ( V n , m ( K ) ) of Grassmann $$G_{n,m}({\mathbb {K}})$$ G n , m ( K ) , flag $$F_{n;n_1,\ldots ,n_k}({\mathbb {K}})$$ F n ; n 1 , … , n k ( K ) and Stiefel $$V_{n,m}({\mathbb {K}})$$ V n , m ( K ) manifolds. Additionally, homotopy nilpotency classes of p-localized $$\Omega (G^+_{n,m}({\mathbb {K}})_{(p)})$$ Ω ( G n , m + ( K ) ( p ) ) and $$\Omega (V_{n,m}({\mathbb {K}})_{(p)})$$ Ω ( V n , m ( K ) ( p ) ) for certain primes p are estimated, where $$G^+_{n,m}({\mathbb {K}})_{(p)}$$ G n , m + ( K ) ( p ) is the oriented Grassmann manifolds. Further, the homotopy nilpotency classes of loop spaces of localized homogeneous spaces given as quotients of exceptional Lie groups are investigated as well.

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Homotopy nilpotency in p-regular loop spaces

Abstract: We consider the problem how far from being homotopy commutative is a loop space having the homotopy type of the p-completion of a product of finite numbers of spheres. We determine the homotopy nilpotency of those loop spaces as an answer to this problem.

Cited by 27 publications

References 29 publications

SAMELSON PRODUCTS IN p-REGULAR SO(2n) AND ITS HOMOTOPY NORMALITY

SAMELSON PRODUCTS IN p-REGULAR SO(2n) AND ITS HOMOTOPY NORMALITY

Homotopy nilpotency in localized $SU(n)$

Homotopy nilpotency of some homogeneous spaces

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