HOMOTOPY NILPOTENCY IN p-REGULAR LOOP SPACES SHIZUO KAJI AND DAISUKE KISHIMOTO

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.

1. Introduction For a prime p, we mean the p-localization and the p-completion in the sense of Bousfield and Kan [4], and denote them by −(p) and −∧ p respectively. We will always assume all spaces have the homotopy types of CW-complexes. H-spaces have been of great interest in algebraic topology and extensively studied. Among other things, the study on the homotopy types of homotopy commutative H-spaces is very successful since we have celebrated Hubbuck’s torus theorem [12]: homotopy commutative connected finite H-spaces are homotopy equivalent to tori. Its mod p analogues are obtained later in various forms by [2], [20], [18] and others. On the other hand, regarding homotopy commutativity, we can consider another problem to determine whether a given H-space is homotopy commutative or not. Now let G be a compact connected Lie group. Mislin [23] showed that the commutator map of G is of finite order in the group of the homotopy set [G × G, G]. This implies that G(p) is homotopy commutative if p is large enough. Motivated by this, McGibbon [19] addressed the question: Question 1.1. For which prime p is the p-localized Lie group homotopy commutative? McGibbon [19] gave the following complete answer to this question when the Lie groups are simple, which also answers the case of simply connected Lie groups. Let G be a compact connected Lie group. It ∏l is classical that the rationalization of G has the homotopy type of the rationalization of i=1 S 2ni −1 with n1 ≤ . . . ≤ nl . In this case, we say that G is of type (n1 , . . . , nl ). Theorem 1.2 (McGibbon [19]). Let G be a compact connected simple Lie group of type (n1 , . . . , nl ). Then we have: (1) G(p) is homotopy commutative if p > 2nl . (2) G(p) is not homotopy commutative if p < 2nl except for (G, p) = (Sp(2), 3), (G2 , 5). This result also holds if we replace the p-localization by the p-completion since it only deals with the p-torsion in the group of the homotopy set [G × G, G] as above (see [23]). It is worth continuing the study on the multiplicative structures of p-completed Lie groups, or, more generally, p-complete loop spaces. Then we consider: Question 1.3. How far from being homotopy commutative is a given p-complete loop space? In order to study Question 1.3, we have to measure a distance from homotopy commutativity. In group theory, if we consider only nilpotent groups, the nilpotency class is the one which measures a distance from commutativity. Then we will adopt the homotopy nilpotency for our purpose which is defined as follows. Let X be a group-like space and let γ1 : X × X → X be the commutator map. Put γn = γ1 ◦ (1 × γn−1 ) for n > 1. Then γn is the n-iterated commutator map of X. X is called homotopy nilpotent of class n if γn is null homotopic but γn−1 is not (see [30]). In this case, we write nilX = n. Then the homotopy nilpotency is the homotopy analogue of the nilpotency of groups. Note that nilX is normalized as nilX = 1 if and only if X is homotopy commutative. It is obvious that, for homotopy nilpotent group-like spaces X1 , . . . , Xk , we have (1)

nil(X1 × · · · × Xn ) = max{nilX1 , . . . , nilXk }.

Of course, not all p-complete loop spaces are homotopy nilpotent but the p-complete loop spaces which we will deal with are all homotopy nilpotent. Then our adoption is plausible. 1

2

SHIZUO KAJI AND DAISUKE KISHIMOTO

Now we specify the class of p-complete loop spaces which we will deal with. Let X be a p-complete loop space. Since we regard Question 1.3 as a generalization of Question 1.1, we should impose a finiteness condition on X such that the mod p cohomology of X is finite. Those loop spaces are now called p-compact groups [7]. We start considering Question 1.3 for relatively large primes since the homotopy types of pcompact groups are simple as follows. Let X be a simply connected p-complete H-space having a finite mod ∏l p cohomology. As is seen above, the rationalization of X has the homotopy type of i=1 K(Q∧ p , 2ni − 1) with n1 ≤ · · · ≤ nl . In this case, we say that X is of type (n1 , . . . , nl ). Of course, this definition of types of p-complete H-spaces is consistent with those of Lie groups above. If X has the homotopy type of the ∏l p-completion of i=1 S 2ni −1 , then we say that X is p-regular. Recall that Kumpel [17] generalized the classical result of Serre [27] as follows. If a simply connected p-complete H-space is of type n1 , . . . , nl and p ≥ nl − n1 + 2, then it is p-regular. Summarizing, the aim of this paper is to consider: Question 1.4. How large is the homotopy nilpotency of a given simply connected p-regular p-compact group ? Remark 1.5. A continuation of Question 1.4 is considered by the second named author [15]. Actually, he considers the homotopy nilpotency of a p-localized SU(n) having the homotopy type of a product of spheres and sphere bundles over spheres. As in the following theorem, one can expect that nilG(p) is a monotonic decreasing function in p for a fixed Lie group G in most cases. But it is shown in [15] that this is false. It is known that p-compact groups have maximal tori and Weyl groups [7], and, for each p-adic pseudoreflection group, there is a connected p-compact group, unique up to isomorphism, having the Weyl group isomorphic to it. We call connected simple p-compact groups which are not the p-completion of Lie groups by exotic p-compact groups (see [1]). Exotic p-compact groups are known to be simply connected and they correspond via Weyl groups to p-adic pseudoreflection groups in Clark and Ewing’s list [5] except for five types when p is odd, and the Dwyer and Wilkerson’s exotic loop space [6] when p = 2. Recently, Andersen, Grodal, Møller and Viruel [1] and Møller [24], [25] gave the classification of p-compact groups in such a way that each connected p-compact group is isomorphic to a product of the p-completion of a Lie group and exotic p-compact groups. Then, by (1), it is sufficient to consider the homotopy nilpotency of p-regular exotic p-compact groups and the p-completion of compact simply connected simple Lie groups which are p-regular in order to answer Question 1.4. Let X be a simply connected p-compact group of type (n1 , . . . , nl ). Wilkerson [29] showed that X is not p-regular for p < nl . Then, by combining with Kumpel’s result [17] above, X is p-regular if and only if p ≥ nl . Then we can list up all p-regular simple p-compact groups and, in particular, five exceptional types of exotic p-compact groups are not p-regular. Now we state the results: Theorem 1.6. Let G be a compact, simply connected, simple Lie group of type (n1 , . . . , nl ). Then we have: (1) nilG(p) = 3 if nl ≤ p ≤ 23 nl for (G, p) = (F4 , 17), (E6 , 17), (E8 , 41), (E8 , 43), (SU(2), 2). (2) nilG(p) = 2 if 23 nl < p < 2nl or (G, p) is in the above exceptional case. We refer the number of the p-adic pseudoreflection group W in Clark and Ewing’s list [5] to N (X) if an exotic p-compact group X corresponds to W . Theorem 1.7. Let X be an exotic p-compact group of type (n1 , . . . , nl ). Then we have: (1) nilX = 1 if p > 2nl . (2) nilX = 2 if nl < p < 2nl except for (N (X), p) = (2b, nl + 1), (23, 11), (30, 31). (3) nilX = 3 if (N (X), p) is in the above exceptional case. The organization of this paper is as follows. In section 2, we will decompose the iterated commutator map γn above and, by using the classical result on the odd primary component of the homotopy groups of spheres, we will prove a result analogous to Theorem 1.6 and Theorem 1.7 in a more general setting but including some indeterminacy. In section 3, the above indeterminacy for Lie groups will be fixed in a case by case analysis. The case of the classical groups will be an easy consequence of the result of Bott [3]. We will show a cohomological criterion for a Samelson product being nontrivial. In the case of the exceptional Lie groups, by calculating the action of P 1 , we will use this criterion to fix the indeterminacy. In section 4, we also fix the indeterminacy of exotic p-compact groups analogously to the exceptional Lie groups using the above cohomological criterion.

HOMOTOPY NILPOTENCY IN p-REGULAR LOOP SPACES

3

2. Commutator map and Samelson products We begin with the following easy lemma in which we normalize nilpotency of groups such that a group is nilpotent of class one if and only if it is abelian. Let K be a group generated by x1 , . . . , xl . For x, y ∈ K, we write xy = yxy −1 . For a positive integer n, we denote by Kn the subgroup of K generated ±1 ±1 y by [x±1 i1 , [· · · [xin , xin+1 ] · · · ]] for all 1 ≤ i1 , . . . , in+1 ≤ l and y ∈ K, where [−, −] is the commutator in K. Lemma 2.1. Let K be as above. Then, for any y1 , . . . , yn+1 ∈ K, the commutator [y1 , [· · · [yn , yn+1 ] · · · ]] ±1 ±1 belongs to Kn . In particular, K is nilpotent of class < n if and only if [x±1 i1 , [· · · [xin , xin+1 ] · · · ]] = e for all 1 ≤ i1 , . . . , in+1 ≤ l, where e is unity of K. Proof. For x, y, z ∈ K, we have [x, yz] = [x, y][x, z]y .

(2)

±1 ±1 ±1 Let x = x±1 i1 · · · xia and y = xj1 · · · xjb for 1 ≤ i1 , . . . , ia , j1 , . . . , jb ≤ l. Then it follows that, for all y = y1 · · · ya , z = z1 · · · zb ∈ K,

[y, z] = [y, z1 ][y, z2 · · · zb ]z1 = [y, z1 ][y, z2 ]z1 [y, z3 · · · zb ]z1 z2 = ··· =

b ∏

[y, zi ]z1 ···zi−1

i=1

=

b ∏

[y2 · · · ya , zi ]z1 ···zi−1 y1 [y1 , zi ]z1 ···zi−1

i=1

=

b ∏

[y3 · · · ya , zi ]z1 ···zi−1 y1 y2 [y2 , zi ]z1 ···zi−1 y1 [y1 , zi ]z1 ···zi−1

i=1

= ··· =

b ∏ a ∏

[yj , zi ]z1 ···zi−1 y1 ···yj−1 .

i=1 j=1

Thus, by combining with the formula

−1

[x, y z ] = [xz , y]z , for x, y, z ∈ K, Lemma 2.1 follows from induction on n.

¤

Let X be a group-like space. We denote the composition of a map α : A → X followed by the homotopy inverse of X by −α. The (generalized) Samelson product of α : A → X and β : B → X, denoted 〈α, β〉, is α∧β

γ ¯

the composition A ∧ B −→ X ∧ X → X, where γ¯ is the reduced commutator map of X. The commutator map and Samelson products are, of course, closely related. In particular, we have: Proposition 2.2. Let X be a group-like space such that X = X1 × · · · × Xl as spaces and let ϵk : Xk → X and πk : X → Xk denote the inclusion and the projection respectively. Then nilX < k if ±(ϵi1 ◦ πi1 ) ◦ 〈±ϵj1 , ±(ϵi2 ◦ πi2 ) ◦ 〈· · · ± (ϵik ◦ πik ) ◦ 〈±ϵjk , ±ϵjk+1 〉 · · · 〉〉 = 0 for each 1 ≤ i1 , . . . , ik , j1 , . . . , jk+1 ≤ l. Proof. Consider the group [X k+1 , X] on which the group structure is given by the pointwise multiplication. Denote the commutator in the group [X k+1 , X] by [−, −]. Then, by definition, the commutator [λ1 , [· · · [λk , λk+1 ] · · · ]] of λ1 , . . . , λk+1 ∈ [X k+1 , X] is the composition: ∆

2

λ1 ×···×λk+1

γk

X k+1 −→ X (k+1) −−−−−−−−→ X k+1 −→ X, 2

where ∆ : X k+1 → X (k+1) is the diagonal map and γk : X k+1 → X is the k-iterated commutator map as in the previous section. Let ρi : X k+1 → X denote the i-th projection. Then it follows from (3)

(ρ1 × · · · × ρk+1 ) ◦ ∆ = 1X k+1

that (4)

γk = [ρ1 , [· · · [ρk , ρk+1 ] · · · ]].

4

SHIZUO KAJI AND DAISUKE KISHIMOTO

Put ι = (ϵ1 ◦ π1 ) · · · (ϵn ◦ πn ), where the right hand side is given by the pointwise multiplication. Then ι is a self-homotopy equivalence of X. We also put θi = ϵi ◦ πi . Let us concentrate in the subgroup K of [X k+1 , X] generated by θi ◦ ρj ◦ ι−1 for 1 ≤ i ≤ l and 1 ≤ j ≤ k + 1. Then, for (4), we have γk ∈ K and hence, by applying Lemma 2.1 to the group K, we obtain that γk = 0 if and only if (5)

[±(θi1 ◦ ρj1 ◦ ι−1 ), [· · · [±(θik ◦ ρjk ◦ ι−1 ), ±(θik+1 ◦ ρjk+1 ◦ ι−1 )] · · · ]] = 0

for each 1 ≤ i1 , . . . , ik+1 ≤ l and 1 ≤ j1 , . . . , jk+1 ≤ k + 1. Since ∆ ◦ ι−1 = (ι−1 × · · · × ι−1 ) ◦ ∆ and ι is a homotopy equivalence, (5) is equivalent to [±(θi1 ◦ ρj1 ), [· · · [±(θik ◦ ρjk ), ±(θik+1 ◦ ρjk+1 )] · · · ]] = 0. Thus, for (3), γk = 0 if

γk ◦ (±θi1 × · · · × ±θik+1 ) = 0

for each 1 ≤ i1 , . . . , ik+1 ≤ l. Since X is a group-like space, the induced map [∧k+1 X, X] → [X k+1 , X] from the pinching map X k+1 → ∧k+1 X is monic (see [30, Lemma 1.3.5]). Then it follows that γk = 0 if and only if (6)

〈±θi1 , 〈· · · 〈±θik , ±θik+1 〉 · · · 〉〉 = 0

for each 1 ≤ i1 , . . . , ik+1 ≤ l. By definition, for maps f1 : A1 → A2 , f2 : A2 → X, g1 : B1 → B2 , g2 : B2 → X, we have 〈f2 ◦ f1 , g2 ◦ g1 〉 = 〈f2 , g2 〉 ◦ (f1 ∧ g1 ). Then (6) implies that γk = 0 if and only if (7)

±θj1 ◦ 〈±θi1 , ±θj2 ◦ 〈· · · ± θjk ◦ 〈±θik , ±θik+1 〉 · · · 〉〉 = 0

for all 1 ≤ i1 , . . . , ik+1 , j1 , . . . , jk ≤ l. Moreover, since (πi1 ∧· · ·∧πik+1 )∗ : [Xi1 ∧· · ·∧Xik+1 , X] → [∧k+1 X, X] is injective, (7) holds if and only if ±θj1 ◦ 〈±ϵi1 , ±θj2 ◦ 〈· · · ± θjk ◦ 〈±ϵik , ±ϵik+1 〉 · · · 〉〉 = 0 for all 1 ≤ i1 , . . . , ik+1 , j1 , . . . , jk ≤ l. Therefore the proof is completed.

¤

Let p be an odd prime. Before applying Proposition 2.2 to our case, let us recall from [28] some basic facts on the p-primary component of the homotopy groups of spheres. Proposition 2.3 ([28, Chapter XIII]). Denote the p-primary component of a finitely generated abelian group A by p A. Then we have the following. { Z/p i = 2p − 3 p 2n−1 ∼ (1) π2n−1+i (S )= 0 i ≤ 4p − 7, i ̸= 2p − 3. (2) Let α1 (3) denote a generator of p π2p (S 3 ) and let α1 (n) = Σn−3 α1 (3). Then α1 (2n − 1) is a generator of p π2n+2p−4 (S 2n−1 ). (3) α1 (3) ◦ α1 (2p) ̸= 0 and Σ2 (α1 (3) ◦ α1 (2p)) = 0. Let X be a simply connected p-complete group-like space of type (n1 , . . . , nl ). Put p > nl − n21 + 1. As is noted above, Kumpel [17] showed that X is p regular if p ≥ nl − n1 + 2 and then we can assume 2ni −1 ∧ 2nl −1 ∧ )p . Denote the inclusion (S 2ni −1 )∧ )p X = (S 2n1 −1 )∧ p → X and the projection X → (S p × · · · × (S by ϵi and πi respectively. Then, by Proposition 2.3, we have { aα1 (2ns − 1) if ni + nj = ns + p − 1 (8) πs ◦ 〈ϵi , ϵj 〉 = 0 otherwise, here a ∈ Z/p is possibly 0. Hence it follows also from Proposition 2.3 that πs ◦ 〈ϵi , (ϵt ◦ πt ) ◦ 〈ϵj , ϵk 〉〉 ̸= 0 if and only if { ns = 2, ni + nt = p + 1, nj + nk = nt + p − 1 (9) π2 ◦ 〈ϵi , ϵt 〉 ̸= 0, πt ◦ 〈ϵj , ϵk 〉 ̸= 0. Therefore, by Lemma 2.2, an easy inspection shows: Theorem 2.4. Let X be a simply connected p-complete group-like space of type (n1 , . . . , nl ). Then we have: (1) nilX = 1 if p > 2nl . (2) nilX = 1 or 2 if:

HOMOTOPY NILPOTENCY IN p-REGULAR LOOP SPACES

5

3 2 nl < p < 2nl . nl − n21 + 1 < p

(a) (b) ≤ 2nl and X does not satisfy (9). (3) nilX = 3 if n1 = 2, nl < p ≤ 23 nl and X satisfy (9). The classical result of James and Thomas [14] yields if the above group-like space X is a loop space and nl − n1 + 2 < p < 2nl then X is not homotopy commutative, that is, nilX > 1. Thus we obtain: Corollary 2.5. Let X be a simply connected p-compact group of type (n1 , . . . , nl ) with n1 ≤ · · · ≤ nl . Then we have: (1) nilX = 1 if p > 2nl . (2) nilX = 2 if: (a) 32 nl < p < 2nl . (b) nl − n21 + 1 < p ≤ 2nl and X does not satisfy (9). (3) nilX = 3 if n1 = 2, nl < p ≤ 23 nl and X satisfy (9). In most cases, the above corollary reduces the proof of Theorem 1.6 and Theorem 1.7 to examining (9) case by case when nl < p < 32 nl . Hereafter we will use the following notation. Let X be a p-compact group such that 2nl −1 ∧ X ≅ (S 2n1 −1 )∧ )p . p × · · · × (S 2ni −1 ∧ Then we denote the inclusion (S 2ni −1 )∧ )p by ϵi and πi respectively. p → X and the projection X → (S

3. Lie groups The latter part of Theorem 1.6 immediately follows from a dimensional reason of Corollary 2.5 (see [22] for the types of simple Lie groups). In the case (G, p) = (G2 , 7), (F4 , 13), (E6 , 13), (E7 , 19), (E8 , 31), Hamanaka and Kono [9] showed that π2 ◦ 〈ϵ2 , ϵn 〉 ̸= 0, where n is the largest entry in the type of G, and then, for Corollary 2.5, we obtain nilG∧ p = 3. Thus the remaining cases of (G, p) are listed in the following table and we will check (9) in these cases, where [x] in the table stands for the largest integer less than or equal to x. G p

SU(n) n≤p≤

Sp(n) 3 2n

2n < p < 3n

Spin(n) 2[ n2 ]


3[ n2 ]

E7

E8

23

37

3.1. Classical groups. 3.1.1. SU(n). Recall that the type of SU(n) is (2, 3, . . . , n). Let p be a prime such that p ≥ n. The classical result of Bott [3] shows that if i + j > n, the order of the Samelson product 〈ϵi , ϵj 〉 is a non-zero multiple of ( ) (i + j − 1)! (10) νp , (i − 1)!(j − 1)! where νp (pk q) = pk for (p, q) = 1. ∼ 3 ∼ ̸ 0 and then nilSU(2)∧ It follows from (10) that 〈1SU(2)∧2 , 1SU(2)∧2 〉 = 2 ≥ 2. Since SU(2) = S , π9 (SU (2)) = ∧ Z/9 (see [28]). Then 〈1SU(2)∧2 , 〈1SU(2)∧2 , 1SU(2)∧2 〉〉 = 0 and therefore nilSU(2)2 = 2. Put 2 < n < p ≤ 23 n. Then it also follows from (2.3) and (10) that π2 ◦ 〈ϵn , ϵp−n+1 〉 ̸= 0 and πp−n+1 ◦ 〈ϵn , ϵ2p−2n 〉 ̸= 0, and hence, for (9), π2 ◦ 〈ϵn , (ϵp−n+1 ◦ πp−n+1 ) ◦ 〈ϵn , ϵ2p−2n 〉〉 ̸= 0. Therefore, by Lemma 2.2 and Corollary 2.5, we have obtained nilSU(n)(p) = 3. Put n = p. This is the only one case which is not covered by Corollary 2.5. By an analogous calculation to the above case, one has π2 ◦ 〈ϵp−1 , (ϵ2 ◦ π2 ) ◦ 〈ϵp−1 , ϵ2 〉〉 = ̸ 0 and then nilSU(p)(p) ≥ 3. Recall from [28] that Σ2 : p π2n+2k (S 2n−1 ) → p π2n+2k+2 (S 2n+1 ) is the zero map. Then πs ◦〈±ϵi , ±(ϵt ◦πt )◦〈±ϵj , ±(ϵu ◦πu )◦ 〈±ϵk , ±ϵl 〉〉〉 = 0 for each 1 ≤ i, j, k, l, s, t, u ≤ p and hence it follows from Lemma 2.2 that nilSU(p)(p) = 3. 3.1.2. Sp(n). Recall that the type of Sp(n) is (2, 4, . . . , 2n). Let p > 2n. The result of Bott [3] also shows that if i + j > n, the order of the Samelson product 〈ϵ2i , ϵ2j 〉 is a non-zero multiple of ) ( (2i + 2j − 1)! . νp (2i − 1)!(2j − 1)! Then, by an analogous calculation to the case of SU(n), we can deduce nilSp(n)∧ p = 3 if 2n < p < 3n.

6

SHIZUO KAJI AND DAISUKE KISHIMOTO

3.1.3. Spin(n). Friedlander [8] showed that if p is an odd prime, there exists an equivalence of loop spaces ∧ Spin(2n + 1)∧ p ≅ Sp(n)p .

Then it follows from the above result on Sp(n) that nilSpin(2n + 1)(p) = 3 if 2n < p < 3n. On the other hand, Harris [10] showed that the canonical fiber sequence Spin(2n) → Spin(2n + 1) → S 2n+1 splits after the completion at an odd prime and thus, by Corollary 2.5, nilSpin(2n)∧ p = 3 if 2n < p < 3n. 3.2. Exceptional groups. We first show a way to find a non-trivial Samelson product by using the Steenrod operations which is used in [16] and also in [9]. Let X be a p-regular connected p-compact group of type (n1 , . . . , nl ). Then we can set (11)

H ∗ (BX; Fp ) = Fp [x1 , . . . , xl ], |xi | = 2ni , ϵ∗i (σ(xi )) = u2ni −1 ,

where σ is the cohomology suspension and un is a generator of H n (S n ; Z/p). Lemma 3.1. Let θ be a mod p Steenrod operation. If θxi contains the term axj xk with a ̸= 0 ∈ Fp , then 〈ϵj , ϵk 〉 ̸= 0. Proof. Suppose that 〈ϵj , ϵk 〉 = 0. Let ad : [V, ΩW ] → [ΣV, W ] denote the adjoint congruence. Then the 2nk ∧ Whitehead product [adϵj , adϵk ] = ad〈ϵj , ϵk 〉 = 0 and hence there exists a map f : (S 2nj )∧ )p → BX p ×(S such that the following square diagram is homotopy commutative. 2nk ∧ (S 2nj )∧ )p p ∨ (S

adϵj ∨adϵk

/ BX ∨ BX ∇

ι

² 2nk ∧ )p (S 2nj )∧ p × (S

² / BX,

f

where ι and ∇ are the inclusion and the folding map respectively. It follows from (11) that (adϵm )∗ (xm ) = u2nm for m = 1, . . . , l. Then if θxi contains the terms axj xk with a ̸= 0 ∈ Fp , f ∗ (θxi ) = au2nj × u2nk ̸= 0. On the other hand,

f ∗ (θxi ) = θf ∗ (xi ) = 0 and this is a contradiction. Thus Lemma 3.2 is established.

¤

Corollary 3.2. Let nl < p ≤ 23 nl and let n1 < · · · < nl . Suppose that n1 = 2, P 1 x1 and P 1 xl contain the terms axi xl and bxj xk for a, b ̸= 0 ∈ Fp respectively. Then nilX = 3. Proof. For n1 < . . . < nl and Proposition 2.3, one has πn ◦ 〈ϵi , ϵl 〉 = 0 unless n = 1. On the other hand, it follows from Lemma 3.1 that 〈ϵi , ϵl 〉 ̸= 0 and then π1 ◦ 〈ϵi , ϵl 〉 ̸= 0. Analogously, one can see that πl ◦ 〈ϵj , ϵk 〉 ̸= 0 and then, for (9) and Lemma 2.5, we have established Corollary 3.2. ¤ We next prepare notation for some symmetric polynomials. Denote ck the k-th elementary symmetric function in t1 , . . . , tn for k = 1, . . . , n, that is, n ∏ 1 + c1 + · · · + cn = (1 + ti ). i=1

Define symmetric polynomials pk and sk for k = 1, . . . , n by 1 − p1 + · · · + (−1)n pn =

n ∏

(1 − t2i )

i=1

and 2k sk = t2k 1 + · · · + tn respectively. Then one has Girard’s formula ∑ (i1 + · · · + in − 1)! i1 p1 · · · pinn (12) sk = (−1)k k (−1)i1 +···+in i1 ! · · · in ! i1 +2i2 +···+nin =k

HOMOTOPY NILPOTENCY IN p-REGULAR LOOP SPACES

7

(see [21]). In the canonical way, we will identify ck and pk with the universal k-th Chern class and Pontrjagin class respectively. 3.2.1. E7 with p = 23. Recall that we have the commutative diagram: (13)

Spin(10)

Spin(10)

i1

² E6

i2

² / E7

j

Recall also that the mod 23 cohomology of BSpin(10), BE6 and BE7 are: H ∗ (BSpin(10); F23 ) = F23 [p1 , p2 , p3 , p4 , c5 ], |pi | = 4i, |c5 | = 10, H ∗ (BE6 ; F23 ) = F23 [x2 , x5 , x6 , x8 , x9 , x12 ], |xi | = 2i, H ∗ (BE7 ; F23 ) = F23 [y2 , y6 , y8 , y10 , y12 , y14 , y18 ], |yi | = 2i. In [9], it is shown that xi and yi can be chosen such that: j ∗ (yi ) = xi (i = 2, 6, 8), i∗1 (x2 ) i∗1 (x8 )

j ∗ (y10 ) = x25 , j ∗ (y14 ) = x5 x9 , i∗1 (x5 ) = c5 ,

= p1 ,

i∗1 (x6 ) = −6p3 + p1 p2 ,

= 12p4 + p22 − 21 p21 p2 .

Then, for (13), one has: (14)

i∗2 (y2 ) = p1 ,

i∗2 (y6 ) = −6p3 + p1 p2 ,

i∗2 (y8 ) = 12p4 + p22 − 21 p21 p2 , i∗2 (y10 ) = c45 .

For a dimensional reason, one also has i∗1 (x9 ) = ap21 c5 − bp2 c5 for a, b ∈ F23 and then it follows from (13) and (14) that i∗2 (y14 ) = ap21 c25 − bp2 c25 .

(15)

We can see from the following proposition that (E7 )∧ 23 satisfies the condition of Corollary 3.2 and then = 3. nil(E7 )∧ 23 Proposition 3.3. P 1 y2 contains the term cy6 y18 with c ̸= 0 and P 1 y6 contains the term dy10 y18 + ey14 y14 with d ̸= 0 or e ̸= 0. Proof. Define a ring homomorphism π : F23 [p1 , . . . , p4 , c5 ] → F23 [a2 , . . . , a4 , b5 ]/(a32 , a23 , a24 , b35 , 12a4 + a22 ) by π(p1 ) = 0, π(pi ) = ai (i = 2, 3, 4), π(c5 ) = b5 . Then, for (14) and (15), one has (16)

π(i∗2 (y2 )) = π(i∗2 (y8 )) = π(i∗2 (y62 )) = π(i∗2 (y14 y10 )) = 0.

Put P 1 y2 = cy6 y18 +other terms for c ∈ F23 . Then, for (16) and a degree reason, one has π(i∗2 (P 1 y2 )) = cπ(i∗2 (y6 y18 )). On the other hand, since p1 = s1 and P 1 s1 = 2s12 , Girard’s formula (12) yields that π(i∗2 (P 1 y2 )) = π(P 1 i∗2 (y2 )) = π(P 1 p1 ) = −15a3 a4 b25 ̸= 0. Thus we have c ̸= 0. We define a ring homomorphism π ′ : F23 [p1 , . . . , p4 , c5 ] → F23 [a′2 , a′4 , b′5 ]/((a′2 )3 , (a′4 )2 , (b′5 )5 , 12a′4 + (a′2 )2 ) by

π ′ (pi ) = 0 (i = 1, 3), π ′ (pj ) = a′j (j = 2, 4), π ′ (c5 ) = b′5 . Then, for (14) and (15), we have (17)

π ′ (i∗2 (y2 )) = π ′ (i∗2 (y6 )) = π ′ (i∗2 (y8 )) = 0.

Put P 1 y6 = dy10 y18 + ey14 y14 +other terms for d, e ∈ F23 . Then it follows from (17) and a degree reason that π ′ (i∗2 (P 1 y6 )) = dπ ′ (i∗2 (y10 y18 )) + eπ ′ (i∗2 (y14 y14 )).

8

SHIZUO KAJI AND DAISUKE KISHIMOTO

For Girard’s formula (12), we have: π ′ (P 1 p1 ) = π ′ (P 1 s1 ) = π ′ (2s12 ) = −a′2 (b′5 )4 π ′ (P 1 s3 ) = π ′ (6s14 ) = −9a′4 (b′5 )4 Since s3 = p31 − 3p1 p2 + 3p3 , one has π(P 1 p3 ) = 9a′4 (b′5 )4 and then, for (14), π ′ (P 1 (i∗2 (y6 ))) = π ′ (P 1 (−6p3 + p1 p2 )) = −19a′4 (b′5 )4 ̸= 0. Therefore we have obtained d ̸= 0 or e ̸= 0 and the proof is completed.

¤

3.2.2. E8 with p = 37. Recall that the mod 37 cohomology of BE8 is given by H ∗ (BE8 ; F37 ) = F37 [z2 , z8 , z12 , z14 , z18 , z20 , z24 , z30 ], |zi | = 2i. In order to see the action of P 1 on H ∗ (BE8 ; F37 ), we shall choose suitable zi . Let αi (i = 1, . . . , 8) and α ˜ be respectively the simple roots and the dominant root of E8 as indicated in the following extended Dynkin diagram of E8 (see [22] for details). α1 b

α3 b

α4 b

α5 b

α6 b

α7 b

α8 b

α ˜ s

b α2 Denote the Weyl group of E8 by W . We consider the subgroup K of W which is generated by the reflections corresponding to αi for i = 2, . . . , 8 and α ˜ . Then, by choosing appropriate generators t1 , . . . , t8 ∈ H 2 (BT ; F23 ), one has H ∗ (BT ; F37 )K = F37 [p1 , . . . , p7 , c8 ], where T is a maximal torus of E8 . This is nothing but the cohomology of BSpin(16) in BE8 . Let ϕ be the elements of W corresponding to α1 . Then, by definition, W is generated by K and ϕ. Hence one has H ∗ (BT ; F37 )W = H ∗ (BT ; F37 )ϕ ∩ F37 [p1 , . . . , p7 , c8 ]. On the other hand, the canonical map i : BT → BE8 induces an isomorphism ∼ =

i∗ : H ∗ (BE8 ; F37 ) → H ∗ (BT ; F37 )W . It is shown in [9] that ϕ(c1 ) = −c1 , ϕ(c2 ) = c2 , ϕ(c8 ) = c8 − 41 c1 c7 , ϕ(p1 ) = p1 and ϕ(pi ) ≡ pi + c1 hi

(18)

mod (c21 )

for i = 2, . . . , 7, where h2 = 23 c3 ,

h3 = − 12 (5c5 + c2 c3 ),

h4 = 21 (7c7 + 3c2 c5 − c3 c4 ),

h5 = − 12 (5c2 c7 − 3c3 c6 + c4 c5 ),

h6 = − 21 (5c3 c8 − 3c4 c7 + c5 c6 ), h7 = 12 (3c5 c8 − c6 c7 ). Then it is immediate that i∗ (z2 ) = p1 .

(19)

By a direct calculation, Hamanaka and Kono [9] showed: Proposition 3.4 (Hamanaka and Kono [9]). If d8 ∈ H 16 (BT ; F37 ) and d12 ∈ H 24 (BT ; F37 ) satisfy ϕ(d8 ) ≡ d8 mod (c21 ) and ϕ(d12 ) ≡ d12 mod (c21 , c22 ), then d8 ≡ a˜ z8

mod (p41 ), d12 ≡ b˜ z12

mod (p21 )

for a, b ∈ F37 and z˜8 = 120p4 + 1680c8 + p21 p2 − 36p1 p3 + 10p22 , z˜12 = 60p6 − p1 p2 p3 − 5p1 p5 +

5 3 36 p2

Since z˜2 , z˜8 , z˜12 are algebraically independent, we have:

− 5p2 p4 + 110p2 c8 + 3p23 .

HOMOTOPY NILPOTENCY IN p-REGULAR LOOP SPACES

9

Corollary 3.5 (Hamanaka and Kono [9]). Let z˜8 and z˜12 be as in Proposition 3.4. Then we can choose generators z8 and z12 of H ∗ (BE8 ; F37 ) such that i∗ (z8 ) ≡ z˜8 , i∗ (z12 ) ≡ z˜12

mod (p21 ).

Let us consider the generator z14 . For a dimensional reason, an element of degree 28 in F37 [p1 , . . . , p7 , c8 ] is given by a linear combination ap7 + bp22 p3 + cp2 p5 + dp3 c8 + ep3 p4

mod (p1 )

for a, b, c, d, e ∈ F37 . It is straightforward to check that ϕ(p22 p3 ) ≡ p22 p3 + 6c1 c33 c4 − 12c1 c3 c4 c6 − 10c1 c24 c5 , ϕ(p2 p5 ) ≡ p2 p5 + 3c1 c23 c7 + 23 c1 c3 c25 − c1 c24 c5 , ϕ(p3 c8 ) ≡ p3 c8 − 41 c1 c23 c7 − 52 c1 c5 c8 + 12 c1 c6 c7 , ϕ(p3 p4 ) ≡ p3 p4 − 12 c1 c33 c4 + 72 c1 c23 c7 + c1 c3 c4 c6 + 5c1 c3 c25 − 52 c1 c24 c5 − 5c1 c5 c8 − 7c1 c6 c7

mod (c21 , c2 ).

Then it follows that: Proposition 3.6. If d14 ∈ H 28 (BT ; F37 ) satisfy ϕ(d14 ) ≡ d14 mod (c21 , c2 ), then d14 ≡ a˜ z14 for a ∈ F37 and

mod (p1 )

z˜14 = 480p7 − p22 p3 + 40p2 p5 − 12p3 p4 + 312p3 c8 .

Since z˜2 , z˜8 , z˜12 , z˜14 are algebraically independent, we obtain: Corollary 3.7. Let z˜14 be as in Proposition 3.6. We can choose a generator z14 of H ∗ (BE8 ; F37 ) such that i∗ (z14 ) ≡ z˜14 mod (p1 ). We choose generators z2 , z8 , z12 , z14 of H ∗ (BE8 ; F37 ) as in (19), Corollary 3.5 and Corollary 3.7. For the ∧ following proposition, we can see that (E8 )∧ 37 satisfies the condition of Corollary 3.2 and then nil(E8 )37 = 3. Proposition 3.8. P 1 z2 and P 1 z8 contain the terms az8 z30 and bz20 z24 with a, b ̸= 0 respectively. Proof. Consider the ring homomorphism π : F37 [p1 , . . . , p7 , c8 ] → F37 [a3 , a4 , a7 , b8 ]/(a23 , a24 , a27 , b48 , a3 a4 − 26a3 b8 − 40a7 ) defined by π(pi ) = 0 (i = 1, 2, 5, 6), π(pj ) = aj (j = 3, 4, 7), π(c8 ) = b8 . Then, for (19), Corollary (3.5) and Corollary 3.7, we have π(i∗ (z2 )) = π(i∗ (z12 )) = π(i∗ (z14 )) = 0 and, for a degree reason, we also have π(i∗ (z18 )) = 0. Put P z2 = az8 z30 +other terms. Thus one has 1

π(i∗ (P 1 z2 )) = aπ(i∗ (z8 z30 )). On the other hand, it follows from Girard’s formula (12) and (19) that π(i∗ (P 1 z2 )) = π(P 1 i∗ (z2 )) = π(P 1 p1 ) = π(P 1 s1 ) = π(2s19 ) = 2a4 a7 b28 ̸= 0 and then a ̸= 0. Define a ring homomorphism π ′ : F37 [p1 , . . . , p7 , c8 ] → F37 [a′2 , a′4 , b′8 ]/((a′2 )2 , (a′4 )6 , (b′8 )6 , a4 + 14b8 ) by π ′ (pi ) = 0 (i = 1, 3, 5, 6, 7), π ′ (pj ) = a′j (j = 2, 4), π ′ (c8 ) = b′8 . Then, for (19), Corollary 3.5 and Corollary (3.7), we have 2 π ′ (i∗ (z4 )) = π ′ (i∗ (z8 )) = π ′ (i∗ (z14 )) = π ′ (i∗ (z12 )) = 0.

10

SHIZUO KAJI AND DAISUKE KISHIMOTO

Put P 1 z8 = by20 z24 +other terms. Then we can see that π ′ (i∗ (P 1 z8 )) = bπ ′ (i∗ (z20 z24 )). Let us make a direct calculation of π ′ (i∗ (P 1 z8 )). It follows from Girard’s formula (12) that: π ′ (P 1 s2 ) = π ′ (4s20 ) = −6(b8 )5

π ′ (P 1 s4 ) = π ′ (8s22 ) = 16a′2 (b′8 )5

π ′ (P 1 c8 ) = π ′ (s18 c8 ) = 26a′2 (b′8 )5 Since s2 = p21 − 2p2 and s4 = p41 − 4p21 p2 + 4p1 p3 + 2p22 − 4p4 , one has π ′ (P 1 p2 ) = 3(b′8 )5 , π ′ (P 1 p4 ) = −a′2 (b′8 )5 and then, for Corollary 3.5, π ′ (i∗ (P 1 z8 )) = π ′ (P 1 i∗ (z8 )) = 120π ′ (P 1 p4 ) + 1680π ′ (P 1 c8 ) + 20a′2 π ′ (P 1 p2 ) = −3a′2 (b′8 )5 ̸= 0. Hence we have obtained b ̸= 0 and thus the proof is completed.

¤

4. Exotic p-compact groups By consulting Clark and Ewing’s list [5], we can easily see that Theorem 1.7 follows from a dimensional reason of Corollary 2.5 except for exotic p-compact groups X of type (n1 , . . . , nl ) satisfying (N (X), p) = (2b, nl + 1), (23, 11), (30, 31). Let K be the Weyl group of X. Then the mod p cohomology of X is the invariant ring of K, Fp [t1 , . . . , tl ]K = Fp [x1 , . . . , xl ], in which |ti | = 2 and xi = 2ni . In particular, we have |x1 | = 4. Note that we can apply Corollary 3.2 to the p-compact groups corresponding to K and the condition in Corollary 3.2 is equivalent to: (20)

P 1 x1 = axi xl + other terms, P 1 xl = bxj xk + other terms

for some i, j, k, l and a, b ̸= 0 ∈ Fp , where the action of P 1 comes from the relation P 1 ti = tpi . Then we will show that K satisfies this condition. 4.1. N (X) = 2b. The pseudoreflection group of number 2b in Clark and Ewing’s list [5] is the Coxeter group I2 (n) for n ≥ 3 which corresponds to the following Coxeter diagram, where p ≡ ±1 (n) (see [13]). b

n

b

For p ≡ 1 (n), it is the dihedral group of order 2n, denoted D2n , acting on F2p via the matrices: ( ) ( ) 0 1 ω 0 , , 1 0 0 ω −1 where ω is an n-th root of unity which exists in Fp since p ≡ 1 (n). Then the invariant ring of D2n is (21)

Fp [t1 t2 , tn1 + tn2 ].

Put p = n + 1 and x2 = t1 t2 , xn = tn1 + tn2 . Then we have P 1 x2 = x2 xn and then, by (20), the proof of Theorem 1.7 for (N (X), p) = (2b, n + 1) is completed. 4.2. (N (X), p) = (23, 11). The pseudoreflection group of number 23 in Clark and Ewing’s list [5] is the Coxeter group H3 corresponding to the Coxeter diagram: b

5

b

b

Let P1 be the invariant ring of I2 (5) over F11 . Then it follows from (21) that P1 = F11 [x2 , xn ] such that (22)

P 1 x2 = x2 x25 − 2x62 .

Denote the invariant ring of H3 over F11 by P2 . Then we have P2 = F11 [y2 , y6 , y10 ], |yi | = 2i in which we can set i(y2 ) = x2 by the map i : P2 → P1 induced from the inclusion of the above Coxeter diagrams. Put P 1 y2 = ay2 y10 +other terms for a ∈ F11 . Then it follows from (22) and a degree reason that x2 x25 ≡ P 1 x2 ≡ P 1 i(y2 ) ≡ i(P 1 y2 ) ≡ ai(y2 )i(y10 )

mod (x22 )

HOMOTOPY NILPOTENCY IN p-REGULAR LOOP SPACES

11

and hence a ̸= 0. Therefore the invariant ring of H3 over F11 satisfies (20). 4.3. (N (X), p) = (30, 31). The pseudoreflection group of number 30 in Clark and Ewing list [5] is the Coxeter group H4 corresponding to the Coxeter diagram: b

5

b

b

b

It follows from (21) that the invariant ring of I2 (5) over F31 is given by Q1 = F31 [x2 , x5 ] for x2 = t1 t2 and x5 = t51 + t52 . Then we have (23)

P 1 x2 ≡ x2 x65 , P 1 x5 ≡ 5x75

mod (x22 ).

Let Q2 = F31 [y2 , y6 , y10 ], |yi | = 2i be the invariant ring of H3 over F31 , where we put i(y2 ) = x2 by the 3 canonical by i : Q2 → Q1 as above. Put P 1 y2 = ay2 y10 +other terms for a ∈ F31 . Then, for (23) and a degree reason, we have 3 x2 x65 ≡ P 1 x2 ≡ P 1 i(y2 ) ≡ i(P 1 y2 ) ≡ ai(y2 )i(y10 )

mod (x22 )

and hence a ̸= 0. Thus we can put i(y10 ) = x25 +other terms. Therefore we obtain (24)

3 4 P 1 y2 = y2 y10 , P 1 y10 = 10y10

mod (y22 , y6 ).

Denote by Q3 the invariant ring of H4 over F31 , F31 [z2 , z12 , z20 , z30 ], |zi | = 2i, in which we can set j(z2 ) = y2 by the canonical map j : Q3 → Q2 . Put P 1 z2 = bz2 z30 + cz12 z20 +other terms for b, c ∈ F31 . Then, for (24), we have (25)

3 y2 y10 ≡ P 1 y2 ≡ P 1 j(z2 ) ≡ j(P 1 z2 )

≡ bj(z2 )j(z30 ) + cj(z12 )j(z20 ) mod (y22 , y6 ).

2 +other terms. Put Hence b ̸= 0 or c ̸= 0. Suppose that b = 0. Then, for (25) we can set j(z20 ) = y10 1 P z20 = dz20 z30 +other terms. Thus, for (23) and a degree reason, we have 5 2 20y10 ≡ P 1 y10 ≡ P 1 j(z20 ) ≡ j(P 1 z20 ) ≡ dj(z20 )j(z30 )

mod (y22 , y6 )

and hence d ̸= 0. Summarizing, we have established that P 1 z2 = bz2 z30 + cz12 z20 +other terms with b ̸= 0 or c ̸= 0 such that if b = 0, P 1 z20 = dz20 z30 +other terms with d ̸= 0. This satisfies (20) and now the proof of Theorem 1.7 is completed. References [1] K.K.S. Andersen, J. Grodal, J.M. Møller and A. Viruel, The classification of p-compact groups for p odd, Ann. of Math. 167 (2008), 95-210. [2] J. Aguad´ e and L. Smith, On the mod p torus theorem of John Hubbuck, Math. Z. 191 (1986), no. 2, 325-326. [3] R. Bott, A note on the Samelson products in the classical groups, Comment. Math. Helv. 34 (1960), 249-256. [4] A.K. Bousfield, D.M. Kan, Homotopy Limits, Completions and Localizations, LNM 304 (1972), Springer-Verlag, BerlinNew York. [5] A. Clark and J. Ewing, The realization of polynomial algebras as cohomology rings, Pacific J. Math. 50 (1974), 425-434. [6] W.G. Dwyer and C.W. Wilkerson, A new finite loop space at the prime two, J. Amer. Math. Soc. 6 (1993), 37-64. [7] W.G. Dwyer and C.W. Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. 139 (1994), 395-442. [8] E.M. Friedlander, Exceptional isogenies and the classifying spaces of simple Lie groups, Ann. of Math. 101 (1975), 510-520. [9] H. Hamanaka and A. Kono, A note on Samelson products in exceptional Lie groups, preprint available at http://www.math.kyoto-u.ac.jp/preprint/2008/15kono.pdf. [10] B. Harris, Suspensions and characteristic maps for symmetric spaces, Ann. of Math. 76 (1962), 295-305. [11] Y. Hemmi, Higher homotopy commutativity of H-spaces and the mod p torus theorem, Pacific J. Math. 149 (1991), no. 1, 95-111. [12] J. Hubbuck, On homotopy commutative H-spaces, Topology 8 (1969), 119-126. [13] J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, Cambridge, 1990. [14] I.M. James and E. Thomas, Homotopy-abelian topological groups, Topology 1 (1962), 237-240. [15] D. Kishimoto, Homotopy nilpotency in localized SU(n), to appear in Homology, Homotopy and Applications. ¯ [16] A. Kono and H. Oshima, Commutativity of the group of self-homotopy classes of Lie groups, Bull. London Math. Soc. 36 (2004), 37-52. [17] P.G. Kumpel, Mod p-equivalences of mod p H-spaces, Quart. J. Math. 23 (1972), 173-178. [18] J.P. Lin, Loops of H-spaces with finitely generated cohomology rings, Topology Appl. 60 (1994), no. 2, 131-152. [19] C. McGibbon, Homotopy commutativity in localized groups, Amer. J. Math. 106 (1984), 665-687. [20] C.A. McGibbon, Higher forms of homotopy commutativity and finite loop spaces, Math. Z. 201 (1989), no. 3, 363-374.

12

SHIZUO KAJI AND DAISUKE KISHIMOTO

[21] J.W. Milnor and J.D. Stasheff, Characteristic Classes, Ann. of Math. Studies 76, Princeton Univ. Press, Princeton N.J., 1974. [22] M. Mimura and H. Toda, Topology of Lie Groups I, II, Translations of Math. Monographs 91, AMS Providence, RI, 1991. [23] G. Mislin, Nilpotent groups with finite commutator subgroups, Lecture Notes in Math., 418, Springer, Berlin, 1974, 103-120. [24] J. Møller, N -determined 2-compact groups II, Fund. Math. 196 (2007), no. 1, 1-90. [25] J. Møller, N -determined 2-compact groups I, Fund. Math. 195 (2007), no. 1, 11-84. [26] L. Saumell, Homotopy commutativity of finite loop spaces, Math. Z. 207 (1991), no. 2, 319-334. [27] J.-P. Serre, Groupes d’homotopie et classes des groupes abeliens, Ann. of Math. 58 (1953), 258-294. [28] H. Toda, Composition Methods in Homotopy Groups of Spheres, Ann. of Math. Studies 49, Princeton Univ. Press, Princeton N.J., 1962. [29] C.W. Wilkerson, K-theory operations and mod p loop spaces, Math. Z. 132 (1973), 29-44. [30] A. Zabrodsky, Hopf Spaces, North-Holland Mathematics Studies 22, North-Holland, Amsterdam, 1976. Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka 814-0180, Japan and Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan