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 Outline Introduction 1 Definitions Motive Work Main Theorem 2 Main Theorem Outline of Proof Basic Tools Upper Bound Lower Bound Homotopy Nilpotency

Introduction Definitions Main Theorem Motive Work Definitions Homotopy Nilpotency X : topological group. iterated commutator map γ n : � n +1 X → X . γ 1 = γ : ( x , y ) �→ xyx − 1 y − 1 , γ n = γ ◦ (1 ∧ γ n − 1 ). homotopy nilpotency of X [BG] nil X = min { n | γ n ≃ ∗} . def X : homotopy commutative ⇐ ⇒ nil X = 1. def X : homotopy nilpotent ⇐ ⇒ 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 Definitions Main Theorem Motive Work Definitions Type of Topological Group X has type ( n 1 , n 2 , . . . , n l ) with n 1 ≤ · · · ≤ n l . ⇒ X ≃ (0) S 2 n 1 − 1 × · · · × S 2 n l − 1 . def ⇐ ⇒ X ≃ ( p ) S 2 n 1 − 1 × · · · × S 2 n l − 1 . def X : p -regular ⇐ (Kumpel [Ku]) p ≥ n l − n 1 + 2 ⇒ X : p -regular. Types of simple Lie groups: A l = SU ( l + 1) (2 , 3 , . . . , l + 1) (2 , 6) G 2 B l = Spin (2 l + 1) (2 , 4 , . . . , 2 l ) (2 , 6 , 8 , 12) F 4 C l = Sp ( l ) (2 , 4 , . . . , 2 l ) E 6 (2 , 5 , 6 , 8 , 9 , 12) D l = Spin (2 l ) (2 , 4 , . . . , 2 l − 2 , l ) E 7 (2 , 6 , 8 , 10 , 12 , 14 , 18) E 8 (2 , 8 , 12 , 14 , 18 , 20 , 24 , 30) Homotopy Nilpotency

Introduction Definitions Main Theorem Motive Work McGibbon’s result Homotopy commutativity in localized groups, Amer. J. Math. 106 (1984) Theorem (McGibbon [Mc]) Let G be a compact, simple Lie group of type ( n 1 , . . . , n l ) with n 1 ≤ · · · ≤ n l . If p > 2 n l , then G ( p ) is homotopy commutative. If p < 2 n l , then G ( p ) is not homotopy commutative except for the cases that ( G , p ) = ( Sp (2) , 3), ( G 2 , 5). There are some generalizations of this work: Saumell, L. Homotopy commutativity of finite loop spaces. Math. Z. 207 (1991), no. 2, 319–334. Saumell, L. Higher homotopy commutativity in localized groups. Math. Z. 219 (1995), no. 2, 203–213. Homotopy Nilpotency

Main Theorem Outline of Proof Introduction Basic Tools Main Theorem Upper Bound Lower Bound Main Theorem Notion of Theorem Theorem G : compact, simple Lie group of type ( n 1 , . . . , n l ) with n 1 ≤ · · · ≤ n l . p : regular prime ⇒ G ( p ) : homotopy nilpotent with: nil ( G ( p ) ) = 1 if 2 n l < p . 1 nil ( G ( p ) ) = 2 if 3 2 n l < p < 2 n l . 2 nil ( G ( p ) ) = 2 if ( G , p ) = ( SU (2) , 2), ( F 4 , 17), ( E 6 , 17), ( E 8 , 41), 3 ( E 8 , 43). nil ( G ( p ) ) = 3 if n l ≤ p ≤ 3 2 n l and ( G , p ) is not the case above. 4 Homotopy Nilpotency

Main Theorem Outline of Proof Introduction Basic Tools Main Theorem Upper Bound Lower Bound Proof Outline of Proof For the upper bounds of homotopy nilpotency: Decompose the commutator by elementary group theory. p -primary part of π ∗ ( S 2 i − 1 ) [To]. and easy arithmetics. For the lower bounds: Non-commutativity Theorem of James and Thomas [JT]. n l < p < 2 n l ⇒ 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

Main Theorem Outline of Proof Introduction Basic Tools Main Theorem Upper Bound Lower Bound Basic Tools Elementary Group Theory H : group generated by x 1 , . . . , x n . [ a , b ] = aba − 1 b − 1 . ( a , b ∈ H ) Z 0 = { x ± 1 | 1 ≤ i ≤ n } , Z k = { [ a , b ] | a ∈ Z 0 , b ∈ Z k − 1 } . i Lower central series H = H 0 ⊃ H 1 = [ H , H ] ⊃ · · · ⊃ H i = [ H , H i − 1 ] ⊃ · · · . Lemma H k is generated by � ∞ i = k Z i . ⇒ we only have to care about commutators for generators. Homotopy Nilpotency

Main Theorem Outline of Proof Introduction Basic Tools Main Theorem Upper Bound Lower Bound Basic Tools 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 α ∧ β γ A ∧ B − → G ∧ G → G . By Definition, nil G ( p ) < k ⇔ � 1 G ( p ) , � 1 G ( p ) , · · · � 1 G ( p ) , 1 G ( p ) � · · · �� ≃ ∗ � �� � k Homotopy Nilpotency

Main Theorem Outline of Proof Introduction Basic Tools Main Theorem Upper Bound Lower Bound Basic Tools Samelson Product G : type ( n 1 , . . . , n l ) with n 1 ≤ · · · ≤ n l . × · · · × S 2 n l − 1 p : regular prime ( ⇒ we regard G ( p ) = S 2 n 1 − 1 ) ( p ) ( p ) ǫ n i : S 2 n i − 1 → G ( p ) : generator. ǫ ′ n i : G → G : π ni ǫ ni S 2 n 1 − 1 × · · · × S 2 n l − 1 → S 2 n i − 1 → S 2 n 1 − 1 × · · · × S 2 n l − 1 − − − ( p ) ( p ) ( p ) ( p ) ( p ) Using the elementary group theory recalled in previous Lemma, Lemma nil G ( p ) < k ⇔ ǫ ′ n j 1 � ǫ n i 1 , ǫ ′ n j 2 � ǫ n i 2 , · · · , ǫ ′ n jk � ǫ n ik , ǫ n ik +1 � · · · �� = 0, 1 ≤ ∀ i m , ∀ j m ≤ l ⇒ We have only to consider maps between odd spheres. Homotopy Nilpotency

Main Theorem Outline of Proof Introduction Basic Tools Main Theorem Upper Bound Lower Bound Upper Bound p -primary components of homotopy groups of spheres From now on, we assume p > n l 1 . We recall some facts on p -primary components of homotopy groups of spheres [To]. � Z / p k = 2 p − 3 π 2 n − 1+ k ( S 2 n − 1 ) = ( p ) 0 0 < k < 4 p − 6 , k � = 2 p − 3 α 1 (3) ∈ π 2 p ( S 3 ( p ) ) = Z / p : generator def = Σ 2 n − 4 α 1 (3) α 1 ( n ) ⇒ π 2 n +2 p − 4 ( S 2 n − 1 ) = Z / p is generated by α 1 (2 n − 1) . ( p ) α 1 (3) ◦ α 1 (2 p ) � = 0. α 1 (2 n − 1) ◦ α 1 (2 n + 2 p − 4) = 0, ( n > 2) 1 For the case p = n l (this occurs only when G = SU ( p )), we have to consider a little more facts on α 2 . Homotopy Nilpotency

Main Theorem Outline of Proof Introduction Basic Tools Main Theorem Upper Bound Lower Bound Upper Bound Samelson products of p -regular groups Recall that ǫ n i and π n j are defined as π nj ǫ ni → S 2 n j − 1 S 2 n i − 1 → S 2 n 1 − 1 × · · · × S 2 n l − 1 − − − ( p ) ( p ) ( p ) ( p ) Since G (0) is homotopy commutative, � ǫ n i , ǫ n j � ∈ π 2( n i + n j − 1) ( G ) : torsion ( N α 1 (2 n s − 1) n i + n j = n s + p − 1 π n s ◦ � ǫ n i , ǫ n j � = n i + n j � = n s + p − 1. @ 0 p > 2 n l ⇒ nil G ( p ) < 2. Since � ǫ n i , ǫ n j ◦ α 1 (2 n j − 1) � = � ǫ n i , ǫ n j � ◦ Σ 2 n i − 1 α 1 (2 n j − 1), There is a non-trivial 3-fold commutator � ǫ n k , � ǫ n i , ǫ n j �� . ⇔ There exist n t and n s such that π n t ◦ � ǫ n k , ǫ n s ◦ π n s ◦ � ǫ n i , ǫ n j �� � = 0 ⇒ n t = 2 , n i + n j = n s + p − 1 , n i + n j + n k = 2 p . @ Therefore p > 3 2 n l ⇒ nil G ( p ) < 3. Trivially nil G ( p ) < 4. Homotopy Nilpotency

Main Theorem Outline of Proof Introduction Basic Tools Main Theorem Upper Bound Lower Bound Upper Bound Case: ( G , p ) = ( SU (2) , 2), ( F 4 , 17), ( E 6 , 17), ( E 8 , 41) There are some exceptional cases in the Main Theorem. @ For the case G = SU (2) = S 3 , p = 2, The only possible 3-fold commutator is � 1 S 3 , � 1 S 3 , 1 S 3 �� ∈ π 9 ( S 3 ) It is known that π 9 ( S 3 ) = Z / 3 Therefore nil SU (2) (2) = 2 For other cases, we use the previous arithmetics. For example, in the case G = E 8 , p = 43: If non-trivial 3-fold commutator exists, there exists n i , n j , n k such that n i + n j = n s + p − 1 , n i + n j + n k = 2 p = 86. However the largest entry n l = 30 and second n l − 1 = 24 Impossible. Homotopy Nilpotency

Main Theorem Outline of Proof Introduction Basic Tools Main Theorem Upper Bound Lower Bound Lower Bound Samelson products in SU ( n ) We give a lower bound for nil SU ( n ) ( p ) . ǫ i ∈ π 2 i − 1 ( SU ( n )) = Z , ( i = 2 , . . . , n ) : generator ˆ ( i + j − 1)! (Bott [Bo]) The order of � ˆ ǫ i , ˆ ǫ j � is divisible by ( i − 1)!( j − 1)! , ( i + j > n ) This leads to the following: � ǫ n , ǫ p − n � � = 0 � ǫ n , ǫ 2 p − 2 n � � = 0 ( n < p < 3 2 n ) then, � ǫ n , � ǫ n , ǫ 2 p − 2 n �� � = 0 Homotopy Nilpotency

Main Theorem Outline of Proof Introduction Basic Tools Main Theorem Upper Bound Lower Bound Lower Bound Facts on Lie Groups For p : odd prime, below inclusions induces monomorphism on π ∗ when localized at p . Sp ( n ) ֒ → SU (2 n ) (1) Spin (2 n + 1) ֒ → Spin (2 n + 2) . (2) and Theorem by Friedlander [Fr], Spin (2 n + 1) ( p ) ≃ Sp ( n ) ( p ) give us desired result for Sp ( n ) , Spin ( n ). Homotopy Nilpotency