SAMELSON PRODUCTS IN Sp(2) 1. Introduction We calculated certain generalized Samelson products of Sp(2) and give two applications.

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1 SAMELSON PRODUCTS IN Sp(2) HIROAKI HAMANAKA, SHIZUO KAJI, AND AKIRA KONO 1. Introduction We calculated certain generalized Samelson products of Sp(2) and give two applications. One is the classification of homotopy types of gauge groups. Let G be a compact Lie group, π : P B a principal G-bundle over a finite complex B. We denote by G(P ), the group of G-equivariant self maps covering the identity map of B. G(P ) is called the (topological) gauge group of P. In [CS] M.Crabb and W.Sutherland prove as P ranges over all principal G-bundles over B, the number of homotopy types of G(P ) is finite if B is connected and G is a compact connected Lie group. In some situations, exact number of homotopy types are calculated ([K], [HK2]). In this paper we show the following: Theorem 1.1. Denote by ɛ 7 a generator of π 7 (Sp(2)) = Z and by G k the gauge group of principal Sp(2) bundle over S 8 classified by kɛ 7. Then G k G k if and only if (140, k) = (140, k ). 3. The other application is on the homotopy commutativity of Sp(2) localized at Theorem 1.2 (cf. [M]). Sp(2) (3) is homotopy commutative. 2. Notation Here we give some notation and facts which we use throughout this note. We use the same symbol for the inclusion Sp(n) U(2n) U(2n + 1), the complexifications BSp( ) BU( ) and BSp(n) BU(2n + 1). Let W n = U( )/U(n), X n = Sp( )/Sp(n), and : X n W 2n+1. Then we have the following commutative diagram of fibration sequences Sp( ) p i X n BSp(n) BSp( ) U( ) p W 2n+1 BU(2n + 1) BU( ) Date: May 8,

2 2 HIROAKI HAMANAKA, SHIZUO KAJI, AND AKIRA KONO Let σ be the cohomology suspension. Those facts listed below are well known: H (BU( )) = Z[c 1, c 2,...] H (BSp( )) = Z[q 1, q 2,...] H (W 2n+1 ) = (x 4n+3, x 4n+5,...) H (X n ) = (y 4n+3, y 4n+7,...) (c2j ) = ( 1) j q j, (c2j+1 ) = 0 p (x 4n+2j 1) = σ(c 2n+j ) = x 4n+2j 1 p (y 4n+4j 1 ) = σ(q n+j ) = y 4n+4j 1 (x 4n+4j 1) = ( 1) n+j y 4n+4j 1, (x 4n+4j 3) = 0 Let a 4n+2j = σ(x 4n+2j+1 ), b 4n+4j 2 = σ(y 4n+4j 1 ) so that H (ΩW 2n+1 ) = Z{a 4n+2,..., a 8n+2 }, ( 8n + 2) H (ΩX n ) = Z{b 4n+2,..., b 8n+3 }, ( 8n + 3). We need the following Lemma which gives information on Ωp. Lemma 2.1. For a map α : Σ 2 X BSp( ), we have ( Ω Ωp σ 2 α ) (a4n+4j 2 ) = (2n + 2j 1)!σ 2 (ch 2n+2j ( (α))) Proof. Use the equality (Ωp) σ(x 4n+4j 1 ) = (2n + 2j 1)!ch 2n+2j in [HK1]. Precisely following the method in [HK1], we have the following Lemma [N]. Lemma 2.2. There is a lift γ of the commutator map γ : Sp(n) Sp(n) Sp(n) such that δ γ = γ where δ = Ωi : ΩX n Sp(n) and γ (b 4n+4k 2 ) = i+j=n+k y 4i 1 y 4j 1. Now we specialize to the case when n = 2. Let A = Sp(2) (7) = S 3 e 7 and ˆɛ : A Sp(2). Define two maps a : ΣA ΣSp(2) Ad(1) BSp(2) BSp( ) and b : ΣA π S 8 Ad(ɛ 7 ) BSp(2) BSp( ), where π is the projection. Then we have (1) (2) ch( (a)) = Σu Σu 7 ch( (b)) = 2Σu 7, where u 3 = ˆɛ (y 3 ) (resp. u 7 = ˆɛ (y 7 )) is a generator of H 3 (A; Z) Z (resp. H 7 (A; Z) Z). Using the short exact sequence 0 = KSp 0 (S 5 ) we have Lemma 2.3. KSp 0 (S 8 ) KSp 0 (ΣA) KSp 0 (S 4 ) KSp 0 (ΣA) = Z Z is generated by a and b. KSp 0 (S 7 ) = 0,

3 SAMELSON PRODUCTS IN Sp(2) 3 3. The order of the Samelson product ɛ 7, 1 Sp(2) [S 7 Sp(2), Sp(2)] First we consider the order of the Samelson product ɛ 7, ˆɛ [S 7 A, Sp(2)]. Lemma 3.1. [Σ 7 A, ΩX 2 ] Z Z. Proof. Recall that dim(σ 7 A) = 14, ΩX 2 = S 10 e 14 e 18. Let F be the homotopy fiber of the inclusion S 11 X 2. Then since 0 = π 15 (S 11 ) π 14 (ΩX 2 ) π 14 (F ) Z is exact, we have π 14 (ΩX 2 ) Z. Apply this to the following exact sequence Z/2 π 11 (ΩX 2 ) π 14 (ΩX 2 ) [Σ 7 A, ΩX 2 ] π 10 (Y ) Z. Definition 3.2. For α [Σ 7 A, ΩX 2 ], define λ(α) = (λ 1 (α), λ 2 (α)) Z Z, where (Ω α) (a 10 ) = λ 1 (α)σ 7 u 3 and (Ω α) (a 14 ) = λ 2 (α)σ 7 u 7. Lemma 3.3. λ : [Σ 7 A, ΩX 2 ] Z Z is a homomorphism and monic. Proof. The map ξ = y 11 y 15 : X 2 K(Z, 11) K(Z, 15) induces a 18 equivalence ξ (0) : (X 2 ) (0) K(Q, 11) K(Q, 15). Since dimσ 7 A = 14, (Ωξ (0) ) : [Σ 7 A, (ΩX 2 ) (0) ] H 10 (ΣA; Q) H 14 (ΣA; Q) is an isomorphism. By the commutative diagram [Σ 7 A, ΩX 2 ] (Ωξ) H 10 (ΣA) H 14 (ΣA) [Σ 7 A, ΩX 2 ] (0) [Σ 7 A, (ΩX 2 ) (0) ] H 10 (ΣA; Q) H 14 (ΣA; Q), (Ωξ (0) ) we have the lemma since [Σ 7 A, ΩX 2 ] is free and λ = (Ωξ). Let D KSp 0 (S 8 ) Z be a generator. Then we have ch( (D)) = v 8, where v 8 is a generator of H 8 (S 8 ; Z). Consider the following diagram S 7 A ɛ 7 ˆɛ Lemma 3.4. γ (ɛ 7 ˆɛ) has order 140. ΩSp( ) Ωp Ω ΩX 2 ΩW 5 γ δ γ Sp(2) Sp(2). Sp(2) Proof. Put γ 1 = γ (ɛ 7 ˆɛ), α 1 = (Ωp ) (σ 2 (D ˆ a)) and β 1 = (Ωp ) (σ 2 (D ˆ b)). Recall that (a 10 ) = b 10, (a 14 ) = b 14 γ (b 10 ) = y 3 y 7 + y 7 y 3 γ (b 14 ) = y 3 y 11 + y 7 y 7 + y 11 y 3 (ɛ 7) (y 7 ) = 12v 7.

4 4 HIROAKI HAMANAKA, SHIZUO KAJI, AND AKIRA KONO Hence we have λ(γ 1 ) = ( 12, 12). By Lemma 2.1, we have λ(α 1 ) = ( 5!, 7! 6 ), λ(β 1) = (0, 2 7!). Since λ is monic and λ(140γ 1 14α 1 + β 1 ) = 0 we get 140γ 1 = 14α 1 β 1. Consider the following exact sequence 0 Im(Ωp ) [Σ 7 A, ΩX 2 ] δ [Σ 7 A, Sp(2)]. This shows that 140γ (ɛ 7 ˆɛ) = 140δ γ 1 = 14δ α 1 δ β 1 = 0. Proposition 3.5. The order of the Samelson product ɛ 7, 1 Sp(2) is 140. Proof. Since the attaching map of the top cell of Sp(2) become trivial after double suspension, there exists a map i : S 17 Σ 7 Sp(2) such that S 17 Σ 7 A i Σ7ˆɛ Σ 7 Sp(2) is a homotopy equivalence. Hence we only have to show that 140γ 2 = 0, where γ 2 = γ (ɛ 7 1 Sp(2) ) i : S 17 Sp(2). Let ɛ 7 π 7 (SU(4)) Z be a generator. Since (ɛ 7) = 2ɛ 7, we have the following commutative diagram Sp(2) Sp(2) γ ɛ 7 1 S 7 2ɛ7 c Sp(2) SU(4) SU(4) γ where γ : SU(4) SU(4) SU(4) is the commutator. Consider the map of fibrations S 3 SU(3) Sp(2) p p SU(4) S 7 Sp(2), SU(4) By the above diagram, we have p γ 2 = p γ 2 = 2p γ (ɛ 7 ). Since p : Z/8 Z/5 π 17 (Sp(2)) π 17 (S 7 ) Z/24 Z/2 induces an injection on 2-primary part by Mimura-Toda [MT], we have 20γ 2 = 0. Proof of Theorem 1.1. By [AB], the classifying space BG(P ) of the gauge group of a principal G-bundle P over a finite complex B, is homotopy equivalent to Map P (B, BG), the connected component of maps from B to BG containing the classifying map of P. Consider the fibre sequence arose from the evaluation fibration G k Sp(2) α k Map kɛ (S8, BSp(2)) Map 7 kɛ 7 (S 8, BSp(2)) e k BSp(2). By Lang [L] Map kɛ 7 (S8, BSp(2)) is homotopy equivalent to Map 0(S 8, BSp(2)) and α k can be identified with 1 Sp(2), kɛ 7 = k 1Sp(2), ɛ 7 in [Sp(2), Map 0(S 8, BSp(2))] = [Σ 8 BSp(2), BSp(2)] = [Σ 7 Sp(2), Sp(2)], where ɛ 7 is the adjoint of ɛ 7 and, denotes the Samelson product. Previous Proposition and the method in [HK2] completes the proof.

5 SAMELSON PRODUCTS IN Sp(2) 5 4. The order of the Samelson product 1 Sp(2), 1 Sp(2) In [M] McGibbon shows that Sp(2) (3) is homotopy commutative. Here we give another proof of this fact. Denote the mod 3 reduction of y 4j+3 (j 2) by the same symbol. Then we have H (X 2 ; Z/3) = (y 11, y 15, y 19,...) and P 1 y 11 = ±y 15. Let E be the homotopy fiber of ρβp 1 u 11 : K(Z (3), 11) K(Z (3), 16), where ρ is mod 3 reduction and u 11 is a generator of H 11 (K(Z (3), 11), Z/3). Since P 1 y 11 = ±y 15 and βp 1 y 11 = 0, the map y 11 : (ΩX 2 ) (3) K(Z (3), 11) lifts to a 17 equivalence f : (ΩX 2 ) (3) E. Since dim(a A) = 14, (Ωf) : [A A, (ΩX 2 ) (3) ] [A A, ΩE] is an isomorphism of groups. Consider the following exact sequence: H 9 (A A; Z (3) ) H 14 (A A; Z (3) ) [A A, ΩE] H 10 (A A, Z (3) ) H 15 (A A; Z (3) ). { 0 k = 9, 15 Since H k (A A; Z (3) ) = we have [A A, ΩE] Z Z (3) k = 10, 14, (3) Z (3). Define λ : [A A, (ΩX 2 ) (3) ] (Z (3) ) 3 by λ(α) = ( λ 1 (α), λ 1(α), λ 2 (α)) where α (Ω ) (a 10 ) = λ 1 (α)u 3 u 7 + λ 1(α)u 7 u 3 and α (Ω ) (a 14 ) = λ 2 (α)u 7 u 7 for α [A A, (ΩX 2 ) (3) ]. Since λ (0) : [A A, (ΩX 2 ) (3) ] (Q) 3 is an isomorphism (see section 3), λ is monic. It is not hard to show : KSp(Σ 2 A A) (3) K(Σ 2 A A) (3) is an isomorphism. Therefore we may consider a ˆ a, a ˆ b + b ˆ a KSp(Σ 2 A A) (3). Put α 1 = 6 5! (Ωp ) (σ 2 (a ˆ a)) and α 2 = 9 2 6! (Ωp ) (σ 2 (a ˆ b + b ˆ a)).then α 1, α 2 [A A, ΩX 2 ] (3). Using equalities ch( (a)) = Σu Σu 7 and ch( (b)) = 2Σu 7, we can easily show λ(α 1 ) = (1, 1, 7), λ(α 2 ) = (3, 3, 42). By the same method as in the proof of Lemma 3.4, we have λ( γ (ˆɛ ˆɛ)) = ( 1, 1, 1). Since λ( γ (ˆɛ ˆɛ) + α (3α 1 α 2 )) = 0 and λ is monic, we have Lemma 4.1. γ (ˆɛ ˆɛ) = 0 in [A A, Sp(2)] (3). Proof of Theorem 1.2. Since ΣA ΣSp(2) has a retraction. Therefore we have to show that the generalized Whitehead product ΣA ΣA BSp(2) vanish. Taking the adjoint, previous Lemma completes the proof. References [AB] M.F.Atiyah and R.Bott, The Yang-Mills equations over Riemann surfaces, Phils. Trans. Ray. Soc. London Ser A., 308 (1982), [CS] M.C.Crabb and W.A.Sutherland, Counting homotopy types of gauge groups, Proc. London. Math. Soc., (3) 81 (2000), [H] H.Hamanaka, On [X, U(n)] when dimx = 2n + 1, J. Math. Kyoto Univ. 44 (2004), no. 3, [HK1] H.Hamanaka and A.Kono, On [X, U(n)] when dimx = 2n, J. Math. Kyoto Univ., 42 (2003) no. 2, [HK2] H.Hamanaka and A.Kono, Unstable K 1 -group and homotopy type of certain gauge groups. Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), no. 1, [K] A.Kono, A note on the homotopy type of certain gauge groups, Proc. Royal Soc. Edinburgh, 117A (1991),

6 6 HIROAKI HAMANAKA, SHIZUO KAJI, AND AKIRA KONO [L] G.E.Lang, The evaluation map and EHP sequence, Pacific J. Math., 44(1973), [M] C.McGibbon, Homotopy commutativity in localized groups, Amer. J. Math. 106 (1984), [MT] M.Mimura and H.Toda, Homotopy groups of symplectic groups, J. Math. Kyoto Univ. 3, [N] T.Nagao, preprint. Department of Natural Science, Hyogo University of Teacher Education, Kato, Hyogo , Japan address: Department of Mathematics, Kyoto University, Kyoto , Japan address: Department of Mathematics, Kyoto University, Kyoto , Japan address: