Homotopy Nilpotency in Localized Lie Groups Shizuo Kaji Daisuke - - PowerPoint PPT Presentation
Introduction Main Theorem Homotopy Nilpotency in Localized Lie Groups Shizuo Kaji Daisuke Kishimoto Department of Mathematics Kyoto University Homotopy Symposium 2006 at Ehime University Homotopy Nilpotency Introduction Main Theorem
Introduction Main Theorem
Shizuo Kaji Daisuke Kishimoto
Department of Mathematics Kyoto University
Homotopy Symposium 2006 at Ehime University
Homotopy Nilpotency
Introduction Main Theorem
1
Introduction Definitions Motive Work
2
Main Theorem Main Theorem Outline of Proof Basic Tools Upper Bound Lower Bound
Homotopy Nilpotency
Introduction Main Theorem Definitions Motive Work
Homotopy Nilpotency
X: topological group. iterated commutator map γn : n+1 X → X. γ1 = γ : (x, y) → xyx−1y −1, γn = γ ◦ (1 ∧ γn−1). homotopy nilpotency of X [BG] nil X = min{n|γn ≃ ∗}. X: homotopy commutative
def
⇐ ⇒ nil X = 1. X: homotopy nilpotent
def
⇐ ⇒ nil X < ∞.
(Hopkins, Rao [Ho],[Ra]) G: compact Lie group G(p) is homotopy nilpotent ⇔ H∗(G; Z) has no p-torsion.
Homotopy Nilpotency
Introduction Main Theorem Definitions Motive Work
Type of Topological Group
X has type (n1, n2, . . . , nl) with n1 ≤ · · · ≤ nl.
def
⇐ ⇒ X ≃(0) S2n1−1 × · · · × S2nl−1. X: p-regular
def
⇐ ⇒ X ≃(p) S2n1−1 × · · · × S2nl−1.
(Kumpel [Ku]) p ≥ nl − n1 + 2 ⇒ X: p-regular.
Types of simple Lie groups:
Al = SU(l + 1) (2, 3, . . . , l + 1) G2 (2, 6) Bl = Spin(2l + 1) (2, 4, . . . , 2l) F4 (2, 6, 8, 12) Cl = Sp(l) (2, 4, . . . , 2l) E6 (2, 5, 6, 8, 9, 12) Dl = Spin(2l) (2, 4, . . . , 2l − 2, l) E7 (2, 6, 8, 10, 12, 14, 18) E8 (2, 8, 12, 14, 18, 20, 24, 30)
Homotopy Nilpotency
Introduction Main Theorem Definitions Motive Work
Homotopy commutativity in localized groups, Amer. J. Math. 106 (1984)
Theorem (McGibbon [Mc]) Let G be a compact, simple Lie group of type (n1, . . . , nl) with n1 ≤ · · · ≤ nl. If p > 2nl, then G(p) is homotopy commutative. If p < 2nl, then G(p) is not homotopy commutative except for the cases that (G, p) = (Sp(2), 3), (G2, 5). There are some generalizations of this work: Saumell, L. Homotopy commutativity of finite loop spaces. Math.
Saumell, L. Higher homotopy commutativity in localized groups.
Homotopy Nilpotency
Introduction Main Theorem Main Theorem Outline of Proof Basic Tools Upper Bound Lower Bound
Notion of Theorem
Theorem G: compact, simple Lie group of type (n1, . . . , nl) with n1 ≤ · · · ≤ nl. p: regular prime ⇒ G(p): homotopy nilpotent with:
1
nil (G(p)) = 1 if 2nl < p.
2
nil (G(p)) = 2 if 3
2nl < p < 2nl.
3
nil (G(p)) = 2 if (G, p) = (SU(2), 2), (F4, 17), (E6, 17), (E8, 41), (E8, 43).
4
nil (G(p)) = 3 if nl ≤ p ≤ 3
2nl and (G, p) is not the case above.
Homotopy Nilpotency
Introduction Main Theorem Main Theorem Outline of Proof Basic Tools Upper Bound Lower Bound
Outline of Proof
For the upper bounds of homotopy nilpotency:
Decompose the commutator by elementary group theory. p-primary part of π∗(S2i−1) [To]. and easy arithmetics.
For the lower bounds:
Non-commutativity Theorem of James and Thomas [JT].
nl < p < 2nl ⇒ nil G(p) > 1.
Bott’s calculation of Samelson product in SU(n) [Bo]. Some facts on classical Lie groups. Finding non-trivial Whitehead product in BG along the method of Hamanaka-Kono [HK].
Homotopy Nilpotency
Introduction Main Theorem Main Theorem Outline of Proof Basic Tools Upper Bound Lower Bound
Elementary Group Theory
H : group generated by x1, . . . , xn. [a, b] = aba−1b−1. (a, b ∈ H) Z0 = {x±1
i
|1 ≤ i ≤ n}, Zk = {[a, b]|a ∈ Z0, b ∈ Zk−1}. Lower central series H = H0 ⊃ H1 = [H, H] ⊃ · · · ⊃ Hi = [H, Hi−1] ⊃ · · · . Lemma Hk is generated by ∞
i=k Zi.
⇒ we only have to care about commutators for generators.
Homotopy Nilpotency
Introduction Main Theorem Main Theorem Outline of Proof Basic Tools Upper Bound Lower Bound
Samelson Product
Definition A, B: space, X: topological group (generalized) Samelson product of maps α : A → G and β : B → G, denoted by α, β, is the composition A ∧ B
α∧β
− → G ∧ G
γ
→ G. By Definition, nil G(p) < k ⇔ 1G(p), 1G(p), · · · 1G(p), 1G(p) · · ·
≃ ∗
Homotopy Nilpotency
Introduction Main Theorem Main Theorem Outline of Proof Basic Tools Upper Bound Lower Bound
Samelson Product
G : type (n1, . . . , nl) with n1 ≤ · · · ≤ nl. p : regular prime (⇒ we regard G(p) = S2n1−1
(p)
× · · · × S2nl −1
(p)
)
ǫni : S2ni−1 → G(p): generator. ǫ′
ni : G → G :
S2n1−1
(p)
× · · · × S2nl−1
(p) πni
− − → S2ni−1
(p) ǫni
− → S2n1−1
(p)
× · · · × S2nl−1
(p)
Using the elementary group theory recalled in previous Lemma, Lemma nil G(p) < k ⇔ ǫ′
nj1 ǫni1 , ǫ′ nj2 ǫni2 , · · · , ǫ′ njk ǫnik , ǫnik+1 · · · = 0,
1 ≤ ∀im, ∀jm ≤ l ⇒ We have only to consider maps between odd spheres.
Homotopy Nilpotency
Introduction Main Theorem Main Theorem Outline of Proof Basic Tools Upper Bound Lower Bound
p-primary components of homotopy groups of spheres
From now on, we assume p > nl 1. We recall some facts on p-primary components of homotopy groups of spheres [To]. π2n−1+k(S2n−1
(p)
) =
k = 2p − 3 0 < k < 4p − 6, k = 2p − 3 α1(3) ∈ π2p(S3
(p)) = Z/p : generator
α1(n)
def
= Σ2n−4α1(3) ⇒ π2n+2p−4(S2n−1
(p)
) = Z/p is generated by α1(2n − 1). α1(3) ◦ α1(2p) = 0. α1(2n − 1) ◦ α1(2n + 2p − 4) = 0, (n > 2)
1For the case p = nl (this occurs only when G = SU(p)), we have to consider a
little more facts on α2.
Homotopy Nilpotency
Introduction Main Theorem Main Theorem Outline of Proof Basic Tools Upper Bound Lower Bound
Samelson products of p-regular groups
Recall that ǫni and πnj are defined as S2ni−1
(p) ǫni
− → S2n1−1
(p)
× · · · × S2nl−1
(p) πnj
− − → S2nj−1
(p)
Since G(0) is homotopy commutative,
ǫni , ǫnj ∈ π2(ni +nj −1)(G) : torsion πns ◦ ǫni , ǫnj = ( Nα1(2ns − 1) ni + nj = ns + p − 1 ni + nj = ns + p − 1.
@
p > 2nl ⇒ nil G(p) < 2.
Since ǫni, ǫnj ◦ α1(2nj − 1) = ǫni, ǫnj ◦ Σ2ni−1α1(2nj − 1),
There is a non-trivial 3-fold commutator ǫnk , ǫni , ǫnj . ⇔ There exist nt and ns such that πnt ◦ ǫnk , ǫns ◦ πns ◦ ǫni , ǫnj = 0 ⇒ nt = 2, ni + nj = ns + p − 1, ni + nj + nk = 2p.
@
Therefore p > 3
2nl ⇒ nil G(p) < 3.
Trivially nil G(p) < 4.
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Introduction Main Theorem Main Theorem Outline of Proof Basic Tools Upper Bound Lower Bound
Case: (G, p) = (SU(2), 2), (F4, 17), (E6, 17), (E8, 41)
There are some exceptional cases in the Main Theorem.
@
For the case G = SU(2) = S3, p = 2, The only possible 3-fold commutator is 1S3, 1S3, 1S3 ∈ π9(S3) It is known that π9(S3) = Z/3 Therefore nil SU(2)(2) = 2 For other cases, we use the previous arithmetics. For example, in the case G = E8, p = 43: If non-trivial 3-fold commutator exists, there exists ni, nj, nk such that ni + nj = ns + p − 1, ni + nj + nk = 2p = 86. However the largest entry nl = 30 and second nl−1 = 24 Impossible.
Homotopy Nilpotency
Introduction Main Theorem Main Theorem Outline of Proof Basic Tools Upper Bound Lower Bound
Samelson products in SU(n)
We give a lower bound for nil SU(n)(p). ˆ ǫi ∈ π2i−1(SU(n)) = Z, (i = 2, . . . , n) : generator (Bott [Bo]) The order of ˆ ǫi, ˆ ǫj is divisible by
(i+j−1)! (i−1)!(j−1)!, (i + j > n)
This leads to the following: ǫn, ǫp−n = 0 ǫn, ǫ2p−2n = 0 (n < p < 3
2n)
then, ǫn, ǫn, ǫ2p−2n = 0
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Facts on Lie Groups
For p: odd prime, below inclusions induces monomorphism on π∗ when localized at p. Sp(n) ֒ → SU(2n) (1) Spin(2n + 1) ֒ → Spin(2n + 2). (2) and Theorem by Friedlander [Fr], Spin(2n + 1)(p) ≃ Sp(n)(p) give us desired result for Sp(n), Spin(n).
Homotopy Nilpotency
Introduction Main Theorem Main Theorem Outline of Proof Basic Tools Upper Bound Lower Bound
P1 and Samelson Product
G: topological group, p: regular prime. ¯ ǫnj : suspension of ǫnj : S2nj−1 ֒ → G(p). xj
def
= hur(¯ ǫnj) ⇒ H∗(BG) = Z/p[x1, . . . , xl] ǫni, ǫnj = 0 ⇔ S2ni ∨ S2nj
¯ ǫni ∨¯ ǫnj
∃θ
BG(p) ⇒ ∀xk, θP1xk = P1θxk = 0 ⇒ ∀xk, the component αxi · xj in P1xk is 0
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Introduction Main Theorem Main Theorem Outline of Proof Basic Tools Upper Bound Lower Bound
Results from Hamanaka-Kono’s paper
We cite the results from the paper by Hamanaka and Kono. By calculating P1 for H∗(BG), they got: Theorem ([HK]) When G is exceptional and p = nl+1, nil G(p) ≥ 3. Therefore, there are only two remaining cases (E7, 23) and (E8, 37).
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Calculation of P1 for H∗(BG)
We need some tedious calculation is necessary. So we just give some strategy for the calculation. For G = E7, we use Spin(10) → E6 → E7. For G = E8, we use Spin(16) → E8. First, write pull-backed generators of H∗(BG) by power-sum on the torus. Calculate P1. With aid of Girard’s formula, express the result by symmetric functions. Then push forward them to H∗(BG).
Homotopy Nilpotency
Introduction Main Theorem Main Theorem Outline of Proof Basic Tools Upper Bound Lower Bound
Homotopy Nilpotency