Homotopy nilpotency of some homogeneous spaces

Let K=R,C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {K}}={\mathbb {R}},\,{\mathbb {C}}$$\end{document}, the field of reals or complex numbers and H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}$$\end{document}, the skew R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}$$\end{document}-algebra of quaternions. We study the homotopy nilpotency of the loop spaces Ω(Gn,m(K))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (G_{n,m}({\mathbb {K}}))$$\end{document}, Ω(Fn;n1,…,nk(K))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (F_{n;n_1,\ldots ,n_k}({\mathbb {K}}))$$\end{document}, and Ω(Vn,m(K))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (V_{n,m}({\mathbb {K}}))$$\end{document} of Grassmann Gn,m(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{n,m}({\mathbb {K}})$$\end{document}, flag Fn;n1,…,nk(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{n;n_1,\ldots ,n_k}({\mathbb {K}})$$\end{document} and Stiefel Vn,m(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{n,m}({\mathbb {K}})$$\end{document} manifolds. Additionally, homotopy nilpotency classes of p-localized Ω(Gn,m+(K)(p))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (G^+_{n,m}({\mathbb {K}})_{(p)})$$\end{document} and Ω(Vn,m(K)(p))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (V_{n,m}({\mathbb {K}})_{(p)})$$\end{document} for certain primes p are estimated, where Gn,m+(K)(p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G^+_{n,m}({\mathbb {K}})_{(p)}$$\end{document} 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.


Introduction
The homotopy nilpotency classes nil X of associative H -spaces X has been extensively studied in addition to their homotopy commutativity. In particular, Hopkins [8] made great progress by giving (co)homological criteria for homotopy associative finite H -spaces to be homotopy nilpotent. For example, he showed that if a homotopy associative finite H -space has no torsion in the integral homology, then it is homotopy nilpotent. Later, Rao [19,20] showed that the converse of the above criterion is true in the case of groups Spin(n) and SO(n) and a connected compact Lie group is homotopy nilpotent if and only if it has no torsion in homology. Eventually, Yagita [26] proved that, when G is a compact, simply connected Lie group, its p-localization G ( p) is homotopy nilpotent if and only if it has no torsion in the integral homology. Although many results on the homotopy nilpotency are known, precise homotopy nilpotence classes have not been determined in most cases.
Let KP n be the projective n-space for K = R, C, the field of reals or complex numbers and H, the skew R-algebra of quaternions. The homotopy nilpotency of the loop spaces (KP n ) has been first studied by Ganea [5], Snaith [22] and then their p-localization ((KP n ) ( p) ) by Meier [15].
For the James reduced product J (X ) of a space X , Cohen and Wu [4] have asked: Question. Is the Cohen group [J (X ), (Y )] nilpotent for any spaces X, Y ?
We were first inspired by the homotopy nilpotency of (HP n ) for any n ≥ 1 which does not appear in the literature known to the author and then by the question above as well. The paper grew out of our desire to develop techniques in calculating homotopy nilpotency classes of loop spaces of homogeneous spaces. Because no answer is known to the question above, we present some homogeneous spaces Y such that the group [J (X ), (Y )] is nilpotent for any space X .
In Section 1, we set stages for developments to come. This introductory section is devoted to a general discussion and establishes notations on homotopy nilpotency of associative H -spaces used in the rest of the paper. Section 2, based on results [27, Chapter II] by Zabrodsky, takes up the systematic study of the homotopy nilpotency of loop spaces (G n,m (K)) and (V n,m (K)) of Grassmann G n,m (K) and Stiefel V n,m (K) manifolds. First, we make use of [27, Lemma 2.6.6] to derive: Then, we obtain the following results: In particular, nil (HP n ) < ∞.
Next, we make use of results [10,11] by Kaji and Kishimoto,[12] by Kishimoto and [14] by McGibbon on p-regular classical groups to estimate the homotopy nilpotency classes of the p-localized spaces (G n,m (K) ( p) ) and (V n,m (K) ( p) ) for K = R, C, the field of reals or complex numbers and H, the skew R-algebra of quaterions.
Section 3 is devoted to the homotopy nilpotency of the loop spaces (OP 2 ) of the Cayley plane OP 2 and of other homogeneous spaces being quotients of exceptional Lie groups. We also make use results [24,Theorem 12.1] by Theriault on the homotopy nilpotency classes of quasip-regular exceptional Lie groups to estimate the homotopy nilpotency classes of p-localized loop spaces of those homogeneeous spaces.

Preliminaries
All spaces and maps in this note are assumed to be connected, based and of the homotopy type of C W -complexes. We also do not distinguish notationally between a continuous map and its homotopy class. We write (X ) (resp. (X )) for the loop (resp. suspension) space on a space X and [Y, X ] for the set of homopoty classes of maps Y → X .
Given a space X , we use the customary notations X ∨ X and X ∧ X for the wedge and the smash product of X , respectively.
Recall that an H -space is a pair (X, μ), where X is a space and μ : X × X → X is a map such that the diagram commutes up to homotopy, where ∇ : X ∨ X → X is the codiagonal map. We call μ a multiplication or an H -structure for X . Two examples of H -spaces come in mind: topological groups and the spaces (X ) of loops on X . In the sequel, we identify an H -space (X, μ) with the space X .
An H -space X is called a group-like space if X satisfies all the axioms of groups up to homotopy. Recall that a homotopy associative an H -space always has a homotopy inverse. More precisely, according to [27, 1.3.2. Corollary] (see also [1,Proposition 8.4.4]), we have: Proposition 1.1. If X is a homotopy associative H -space then X is a group-like space.
If X is a homotopy associative H -space, then the functor [−, X ] takes its values in the category of groups. One may then ask when this functor takes its values in various subcategories of groups.
For example, X is homotopy commutative if and only if [Y, X ] is abelian for all Y .
Given a group-like space X , we write γ X,1 = id X : X → X andγ X,2 : X × X → X for the commutator map of X . Observe that the restriction ofγ X,2 to the wedge X ∨ X is null homotopic soγ X,2 extends to a map γ X,2 : X ∧ X → X.
For an integer n ≥ 1, let X ∧(n+1) be the (n + 1)-fold smash power of X . Define the iterated Samelson product γ X,n+1 : X ∧(n+1) → X inductively by γ X,n+1 = γ X,2 • (γ X,1 ∧ γ X,n ) for n ≥ 2. Notice that γ X,n has a universal property: any Samelson product of length n on X factors through γ X,n . Lemma 1.2. If X is a homotopy associative H -space and Y a finite dimensional C W -complex with dim Y = n then the group [Y, X ] is nilpotent with the nilpotency class at most n.
Proof. First, recall that given a homotopy associative H -space X , in view of [9], all its m-th Postnikov stages P m X are also a homotopy associative H -space and the canonical map X → P m X is an H -map. Hence, for a C W -complex Y with dim Y = n, there is an isomorphism [Y, X ] ≈ [Y, P n X ] determined by the canonical map X → P n X . Then, the map γ P n X,n+1 ( We point out that the result above has been stated in [8,20] but for a finite C W -complex and a finite homotopy associative H -space X , only. Next, any C Wcomplex Y can be expressed as with Y α finite. This leads to the short exact sequence for any connected homotopy associative H -space X .
In One might ask if there is an upper bound for the nilpotency class of [Y, X ] that is independent of Y . The homotopy nilpotency class of X is the least n such that γ X,n+1 * and γ X,n * . In this case, we write nil X = n and call the homotopy associative H -space X homotopy nilpotent.
Note that nil X = 1 if and only if X is homotopy commutative. By [2, Theorem 2.7], one has: for the m-th Cartesian power X m of X . Given a space X , the number nil (X ) (if any) is called the homotopy nilpotency class of X .
The first major advance was made by Hopkins [ This corollary implies: But, the homotopy nilpotency does not imply the nilpotency of a topological group.
for n ≥ 1, we derive that the groups: (1) SO(n) and O(n) are not nilpotent for n ≥ 3; (2) U (n) and SU (n) are not nilpotent for n ≥ 2; (3) Sp(n) is not nilpotent for n ≥ 1.
Later, Rao [19] showed that the converse of the criterion from Corollary 1.5 is true in the case of Spin(n) and SO(n) by showing that Spin(n), SO(n), n ≥ 7 and SO(3), SO (4) are not homotopy nilpotent.
Because of H -homotopy equivalences O(n) SO(n) × Z 2 and U (n) SU (n) × S 1 , we derive: . Then, notice that by Bott periodicity theorem: 8 Let now G ( p) stand for the p-localization in the sense of [3] of a compact Lie group G. Then, Yagita [26,Theorem] Now, write J (X ) for the James reduced product on a space X and BG for the classifying space of a topological group G. In view of [23,Theorem 8.6], there is an H -map G → (BG) which is a weak homotopy equivalence. Consequently, for a Lie group G there is an H -homotopy equivalence (BG) G. Then, for the question [4] by Cohen and Wu, we conclude: Corollary 1.9. If n ≥ 1 and X is a space then:

Main results
Let f : X → Y be an H -map of homotopy associative H -spaces. Recall from [27, Chapter II] that: Notice that nil f ≤ min{nil X, nil Y }. (1) If nil q ≤ n and i : F → E is central then nil E ≤ n + 1; If a topological group G acts freely on a paracompact spaces X then there is a homeomorphism Since the connecting map ∂ X : (X/G) → G, in view of [23,Theorem 8.6] (see also [6,Corollary 3.4]), is an H -map, the fibration X → X/G → EG/G ≈ BG leads to the H -fibration Now, let G be a compact Lie group and K < G its closed subgroup. Then, the quotient space G/K is a manifold and the quotient map q : G → G/K is a submersion. Hence, q : G → G/K has a local section at the point q(e) = K for the unit e ∈ G. This certainly implies that the map q : G → G/K has a local section at any point q(g) for any g ∈ G. Consequently, the quotient map q : G → G/K is a fiber bundle with the fiber K as a principal K -bundle. Thus, we have a K -fibration Since this fibration is induced by the inclusion map K → G which is certainly a K -map, Lemma 2.1 yields: Proposition 2.2. If G is a compact Lie group and K < G its closed subgroup with nil K < ∞ then nil (G/K ) < ∞.

Grassmannians
Let K = R, C be the field of reals or complex numbers and H, the skew R-algebra of quaternions. Then, we set: if K = H. Write G n,m (K) (resp. G + n,m (K)) for the (resp. oriented) Grassmannian of mdimensional subspaces in the n-dimensional K-vector space. For example, the set of lines G n+1,1 (K) = KP n , the projective n-space over K.
It is well known that G n,m (K) (resp. G + n,m (K)) are smooth manifolds with diffeomorphisms Since the homomorphism π 1 (SU (m) K ) → π 1 (SU (n) K ) of fundamental groups determined by the inclusion map SU (m) K → SU (n) K for 2 ≤ m ≤ n is an epimorphism, we derive that the spaces G + n,m (K) are simply connected for K = R, C, H. Next, there is the universal covering map  Then, Meier [15,Theorem 5.4] has shown some results on the homotopy nilpotency of p-localized projective spaces: Theorem 2.6. Let p be an odd prime and n ≥ 2 a natural number. Then: Since the space RP 2n+1 is simple, there is its localization RP 2n+1 for any prime p ≥ 2. It it also easily to see that nil (RP 2n+1 ( p) ) = 1 for any odd prime p and n ≥ 0. But, the nilpotency nil (HP n ) for any n ≥ 2 does not appear in the literature known to the author. Next, recall that the classifying (1) nil (G + n,m (R) ( p) ) < ∞ for p ≥ 3; (2) nil (G n,m (K)) < ∞ and nil (G + n,m (K)) < ∞ for K = C, H. In particular, nil (HP n ) < ∞.
We do not mention above any result on the p-localization of G n,m (R) because we are not sure on its existence.
Next, recall the following result of Hopf [7]. Let X be a connected H -space with dim H * (X ; Q) < ∞ for the field Q of rationals. Then, one has a homotopy equivalence where − (0) means the rationalization. The sequence (n 1 , . . . , n l ) with n 1 ≤ · · · ≤ n l is called the type of G. Wilkerson [25] showed that a simply connected compact group G is not p-regular for p < n l . Then, by combining with Kumpel's result [13] above, G is p-regular if and only if p ≥ n l .
The types of the connected compact simple Lie groups have been summarized in [18, Chapter IV] as follows: Let G be a compact, connected Lie group of type (n 1 , . . . , n l ). By Serre [21], the group G is p-regular for an odd prime p if there is a p-local (or p-complete) homotopy equivalence G S 2n 1 −1 × · · · × S 2n l −1 .

Stiefel manifolds
The Stiefel manifold V n,m (K) is the set of all orthonormal m-frames in the vector space K n . That is, it is the set of ordered orthonormal m-tuples of vectors in K n for K = R, C or H. It is well known that V n,m (K) is a smooth manifold and there are diffeomorphisms: Since the homomorphism π 1 (SU (m) K ) → π 1 (SU (n) K ) of fundamental groups determined by the inclusion map SU (m) K → SU (n) K for 2 ≤ m ≤ n is an epimorphism, we derive that the spaces V n,m (K) are simply connected for K = R, C, H. Then, Corollary 1.5 and Proposition 2.2 lead to: Proposition 2.17. If 1 ≤ m ≤ n then: Then, for the question [4] by Cohen and Wu, we conclude: for K = R, C, H and a prime p, we may state: Corollary 2.19. If 1 ≤ m ≤ n then: In particular, if G be a compact Lie group, K < G its closed subgroup and q : K → G/K the quotient map then the fibration leads to a right homotopy inverse of the loop map (q) : (G ( p) ) → (G/K ) ( p) provided G is p-regular and the canonical map K ( p) → G ( p) has a left homotopy inverse. Since the localized odd sphere S 2n+1 ( p) is an associative H -space for p > 3, Lemma 3.4 implies that space ((G/K ) ( p) ) is homotopy commutative or equivalently nil ((G/K ) ( p) ) = 1.
Note that the p = 5 case in part (4) is quasip-regular, and the p = 5, 7, 11 cases in part (5) are quasip-regular. The others are all p-regular.
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