Odd primary homotopy types of the gauge groups of exceptional Lie groups
Sho Hasui, Daisuke Kishimoto, Tseleung So, Stephen Theriault
aa r X i v : . [ m a t h . A T ] M a r ODD PRIMARY HOMOTOPY TYPES OF THE GAUGE GROUPS OFEXCEPTIONAL LIE GROUPS
SHO HASUI, DAISUKE KISHIMOTO, TSELEUNG SO, AND STEPHEN THERIAULT
Abstract.
The p -local homotopy types of gauge groups of principal G -bundles over S areclassified when G is a compact connected exceptional Lie group without p -torsion in homologyexcept for ( G, p ) = (E , Introduction
Let G be a topological group and P be a principal G -bundle over a base X . The gaugegroup G ( P ) is the topological group of G -equivariant self-maps of P covering the identity mapof X . As P ranges over all principal G -bundles over X , we get a collection of gauge groups G ( P ). In [13], Kono classified the homotopy types in the collection of G ( P ) when G = SU(2)and X = S . Since then, the homotopy theory of gauge groups has been deeply developed inconnection with mapping spaces and fiberwise homotopy theory incorporating higher homotopystructures. The classification of the homotopy types has been its motivation and considerableprogress has been made towards it.Suppose that G is a compact connected simple Lie group. Then there is a one-to-one cor-respondence between principal G -bundles over S and π ( G ) ∼ = Z . Let G k denote the gaugegroup of the principal G -bundle over S corresponding to k ∈ Z . Most of the classification ofthe homotopy types of gauge groups has been done in this setting, generalizing the situationstudied by Kono [2, 4, 7, 8, 12, 19, 20]. As we will see, the homotopy type of G k is closelyrelated with a certain Samelson product in G . It is shown in [11] that if p is a prime large withrespect to the rank of G , then the classification of the p -local homotopy types of G k reducesto determining the order of this Samelson product. So in [10, 21], the p -local classification isobtained by calculating this Samelson product when G is a classical group.However, when G is exceptional, the classification, up to one factor of 2, for G = G [12] isthe only one result obtained so far. The aim of this paper is to classify the p -local homotopytypes of G k when G is exceptional and has no p -torsion in H ∗ ( G ; Z ). Recall that exceptionalLie groups consist of five types G , F , E , E , E and they have no p -torsion in H ∗ ( G ; Z ) if and Mathematics Subject Classification.
Primary 55P15, Secondary 54C35.
Key words and phrases. gauge group, homotopy type, exceptional Lie group, Samelson product. only if(1.1) G , p ≥
3; F , E , E , p ≥
5; E , p ≥ . All but one case is included in our results. We define an integer γ ( G ) by: γ (G ) = 3 · γ (F ) = 5 · γ (E ) = 5 · · γ (E ) = 7 · · γ (E ) = 7 · · · · m and n , let ( m, n ) denote the greatest common divisor of m and n , and for aninteger k = p r q with ( p, q ) = 1, let ν p ( k ) = p r denote the p -component of k . Now we state ourmain result. Theorem 1.1.
Let G be a compact connected exceptional simple Lie group without p -torsionin H ∗ ( G ; Z ) except for ( G, p ) = (E , . Then G k and G l are p -locally homotopy equivalent ifand only if ν p (( k, γ ( G ))) = ν p (( l, γ ( G ))) . Gauge groups and Samelson products
We first recall a relation between gauge groups and Samelson products. Let G be a compactconnected simple Lie group and let ǫ be a generator of π ( G ) ∼ = Z . Then by definition G k isthe gauge group of a principal G -bundle over S corresponding to kǫ . Let map( X, Y ; f ) denotethe connected component of the mapping space map( X, Y ) containing the map f : X → Y . Itis proved by Gottlieb [3] (cf. [1]) that there is a homotopy equivalence B G k ≃ map( X, BG ; k ¯ ǫ ) , where ¯ ǫ : S → BG is the adjoint of ǫ : S → G . Then by evaluating at the basepoint of S , weget a homotopy fibration sequence G k → G ∂ k −→ Ω G → B G k → BG, so G k is homotopy equivalent to the homotopy fiber of the map ∂ k . Thus we must identify themap ∂ k , which was done by Lang [14]. Lemma 2.1.
The adjoint S ∧ G → G of the map ∂ k is the Samelson product h kǫ, G i . By the linearity of Samelson products, we have:
Corollary 2.2. ∂ k ≃ k ◦ ∂ . Hereafter we localize spaces and maps at a prime p . We next recall the result of Kono,Tsutaya and the second author [11] on the p -local homotopy types of G k . An H-space X iscalled rectractile if the following conditions are satisfied:(1) there is a space A such that H ∗ ( X ) = Λ( e H ∗ ( A ));(2) there is a map λ : A → X which is the inclusion of the generating set in homology. DD PRIMARY HOMOTOPY TYPES OF THE GAUGE GROUPS OF EXCEPTIONAL LIE GROUPS 3 (3) Σ λ has a left homotopy inverse.The space A is often called a homology generating subspace. By [18] it is known that acompact connected simple Lie group G is rectractile at the prime p if ( G, p ) is in the followingtable. SU( n ) ( p − + 1 ≥ n G p ≥ n ) , Spin(2 n + 1) ( p − + 1 ≥ n F , E p ≥ n ) ( p − + 1 ≥ n −
1) E , E p ≥ p -torsion in H ∗ ( G ; Z ) if and only if G is rectractile at theprime p , except for ( G, p ) = (G , , (E , γ ( G, p ) be the order of the Samelson product h ǫ, λ i . In [11] the following reduction of the classification of the p -local homotopy types of G k is proved. Theorem 2.3.
Let G be a compact connected simple Lie group which is rectractile at the prime p except for ( G, p ) = (G , . Then G k and G l are p -locally homotopy equivalent if and only if ( k, γ ( G, p )) = ( l, γ ( G, p )) .Remark . The range of p in [11] is smaller than the above statement because the range in [11]was taken so that G is rectractile and the mod p decomposition satisfies a certain universality.However, we can easily see that only the retractile property was used in [11], so we can improvethe range of p as above.We set notation for G . By the classical result of Hopf, any connected finite H-space isrationally homotopy equivalent to the product of odd dimensional spheres S n − × · · · × S n l − for n ≤ · · · ≤ n l . The sequence ( n , · · · , n l ) is called the type of X . Hereafter, we assume that G is of type ( n , · · · , n l ), without p -torsion, and rectractile at p .We next decompose the Samelson product h ǫ, λ i in G into small pieces. Recall that since weare localizing at p , G decomposes as G ≃ B × · · · × B p − where the type of B i consists of the integers n in the type of G with 2 n − i + 1modulo 2( p −
1) [15]. This is called the mod p decomposition of G . If the rank of B i is lessthan p , then there is a subspace A i of B i such that B i is rectractile with respect to the inclusion A i → B i . From this, one can deduce the retractile property for G as above, so we may take thespace A as A = A ∨ · · · ∨ A p − . Then it follows that the Samelson product h ǫ, λ i is the directsum of h ǫ, λ i , . . . , h ǫ, λ p − i , where λ i : A i → G is the restriction of λ to A i . Let γ i ( G, p ) be theorder of h ǫ, λ i i . Then we have: SHO HASUI, DAISUKE KISHIMOTO, TSELEUNG SO, AND STEPHEN THERIAULT
Proposition 2.5. If G is rectractile at the prime p , then γ ( G, p ) = max { γ ( G, p ) , . . . , γ p − ( G, p ) } . Thus if we calculate γ i ( G, p ) for 1 ≤ i ≤ p − G, p ) then by Proposi-tion 2.5 we know γ ( G, p ) and by Theorem 2.3 we can describe the p -local homotopy types ofthe corresponding gauge groups. These calculations are carried out in the subsequent sections.The following table lists explicit mod p decompositions of exceptional Lie groups without p -torsion in homology except for G , where B (2 m − , . . . , m k −
1) is an H-space of type( m , . . . , m k ). One can easily deduce the types of exceptional Lie groups from this table.F p = 5 B (3 , × B (15 , p = 7 B (3 , × B (11 , p = 11 B (3 , × S × S p ≥ S × S × S × S E p = 5 F × B (9 , p ≥ × S × S E p = 5 B (3 , , , , × B (15 , p = 7 B (3 , , × B (11 , , × S p = 11 B (3 , × B (15 , × S × S × S p = 13 B (3 , × B (11 , × S × S × S p = 17 B (3 , × S × S × S × S × S p ≥ S × S × S × S × S × S × S E p = 7 B (3 , , , × B (23 , , , p = 11 B (3 , × B (15 , × B (27 , × B (39 , p = 13 B (3 , × B (15 , × B (23 , × B (35 , p = 17 B (3 , × B (15 , × B (27 , × S × S p = 19 B (3 , × B (23 , × S × S × S × S p = 23 B (3 , × B (15 , × S × S × S × S p = 29 B (3 , × S × S × S × S × S × S p ≥ S × S × S × S × S × S × S × S If each B i is a sphere, then we call G p -regular . If each B i is a sphere or a rank 2 H-space B (2 m − , m + 2 p − G quasi- p -regular . In the following we calculate γ ( G, p )for different cases. 3.
The p -regular case Throughout this section, we let (
G, p ) be(3.1) F , E , p ≥
13; E , p ≥
19; E , p ≥ . DD PRIMARY HOMOTOPY TYPES OF THE GAUGE GROUPS OF EXCEPTIONAL LIE GROUPS 5
By the above table, this is equivalent to the case when G is p -locally homotopy equivalent toa product of spheres, and such a G is called p -regular. This is also equivalent to p ≥ n l + 1.The homotopy groups of p -localized spheres are known for low dimension. Theorem 3.1 (Toda [22]) . Localized at p , we have π n − k ( S n − ) ∼ = Z /p Z for k = 2 i ( p − − , ≤ i ≤ p − Z /p Z for k = 2 i ( p − − , n ≤ i ≤ p − other cases for ≤ k ≤ p ( p − − , By Theorem 3.1 we have p · π k ( S n − ) = 0 for 0 ≤ k ≤ n + p ( p − −
2, so p · π k ( G ) = 0for 0 ≤ k ≤ p ( p − h ǫ, λ i i is a map from a 2 n -dimensionalsphere into G for n ≤ n l + 1 ≤ p , we get: Proposition 3.2. If G is p -regular, then γ i ( G, p ) ≤ p for any i . On the other hand, the non-triviality of the Samelson product h λ i , λ j i is completely de-termined in [5], where λ i is the inclusion S n i − → G of the sphere factor in the mod p decomposition of G . Theorem 3.3.
The Samelson product h λ i , λ j i is non-trivial if and only if n i + n j = n k + p − for some n k in the type of G . Then we immediately get:
Corollary 3.4. (1)
The Samelson product h ǫ, λ l i is non-trivial for p = n l + 1 ; (2) The Samelson product h ǫ, λ i i is trivial for any i for p > n l + 1 . Thus by combining Propositions 2.5 and 3.2 and Corollary 3.4, we obtain the following.
Corollary 3.5.
We have γ ( G, p ) = ( p p = n l + 11 p > n l + 1 . The quasi- p -regular case Throughout this section, we assume that (
G, p ) is(4.1) F , E , p = 5 , ,
11; E , p = 11 , ,
17; E , p = 11 , , , , , G being a product of spheres and rank 2 H-spaces,and such a G is quasi- p -regular.We first show the triviality of h ǫ, λ i i from the homotopy groups of G . By Theorem 3.1,for n ≥ i ≤ n + ( p − , we have π i ( S n +1 ) = 0 unless i ≡ n mod ( p −
1) and p · π i ( S n +1 ) = 0. SHO HASUI, DAISUKE KISHIMOTO, TSELEUNG SO, AND STEPHEN THERIAULT
Moreover, the homotopy groups of the H-spaces B (2 n − , n − p − p decomposition of G are calculated. Theorem 4.1 (Mimura and Toda [16], Kishimoto [9]) . Localized at p , we have π n − k ( B (3 , p + 1)) ∼ = Z /p Z for k = 2 i ( p − − , ≤ i ≤ p − Z for k = 2 p − other cases for ≤ k ≤ p ( p − − , and π n − k ( B (2 n − , n + 2 p − ∼ = Z /p Z for k = 2 i ( p − − , ≤ i ≤ p − Z /p Z for k = 2 i ( p − − , n ≤ i ≤ p − Z for k = 2 p − other cases for ≤ k ≤ p ( p − − . For n ≥ i ≤ n + ( p − , π i ( B (2 n + 1 , n + 1 + 2( p − i ≡ n mod ( p − p · π i ( B (3 , p + 1)) = 0 for i ≤ p − . By Theorems 3.1 and 4.1, we obtain thefollowing table for all possibly non-trivial Samelson products h ǫ, λ i i in G : p = 5 p = 7 p = 11 p = 13 p=19F i = 3 i = 5E i = 3 i = 2 , i = 3 , i = 5E i = 9 i = 5 , i = 5 , p -localized exceptional Lie group G for ( G, p ) in (1.1) except for (
G, p ) = (G , , (E , G = E , E , E since the F case can be deduced from the E case. Consider the 27-, 56- and 248-dimensional irreducible representations of G = E , E , E ,respectively. Let ρ : G → SU( ∞ ) be the composite of these representations and the inclusionsinto SU( ∞ ). Then we have a homotopy fibration sequenceΩSU( ∞ ) δ −→ F → G ρ −→ SU( ∞ )in which all maps are loop maps, where F is the homotopy fiber of the map ρ . For any finiteCW complex Z , we obtain an exact sequence of groups e K ( Z ) ∼ = e K − ( Z ) δ ∗ −→ [ Z, F ] → [ Z, G ] → e K − ( Z ) . In particular, if Z has only even dimensional cells, then e K − ( Z ) = 0 and we get(4.2) [ Z, G ] ∼ = [ Z, F ] / Im δ ∗ . DD PRIMARY HOMOTOPY TYPES OF THE GAUGE GROUPS OF EXCEPTIONAL LIE GROUPS 7
In [6], the right hand side of (4.2) is identified by using the cohomology of Z under someconditions and we recall it in the special case Z = S ∧ A i . Let SU( ∞ ) ≃ C × · · · × C p − bethe mod p decomposition of SU( ∞ ) where C i is of type ( i + 1 , i + p, · · · , i + 1 + k ( p − , · · · ). Theorem 4.2.
Let ( G, p ) be as in (1.1) except for G = G and ( G, p ) = (E , . For given ≤ i ≤ p − , we suppose the following conditions: (1) for ≤ j ≤ p − with j ≡ i + 2 mod ( p − , B j = B (2 j + 1 , j + 1 + 2( p − , . . . , j + 1 + 2 r ( p − the composite B j incl −−→ G ρ −→ SU( ∞ ) proj −−→ C j is surjective in cohomology; (3) dim A i ≤ j − r + 2)( p − .Then for n = j + ( r + 1)( p − , there is an injective homomorphism Φ : [ S ∧ A i , F ] → H n ( S ∧ A i ; Z ( p ) ) ⊕ H n +2( p − ( S ∧ A i ; Z ( p ) ) such that Φ( δ ∗ ( ξ )) = ( n !ch n ( ξ ) , ( n + p − n + p − ( ξ )) for ξ ∈ e K ( S ∧ A i ) , where ch k is the k -dimensional part of the Chern character.Proof. To better understand the statement of the theorem, we sketch the proof in [6]. Let ρ j be the composition B j incl −→ G ρ −→ SU ( ∞ ) proj −→ C j and let F j be its homotopy fiber. Byhypothesis, ˜ H i ( F j ; Z ( p ) ) is zero for i ≤ j + 2( r + 2)( p − H n ( F j ; Z ( p ) ) and˜ H n +2( p − ( F j ; Z ( p ) ) which are Z ( p ) . Moreover, the generators a n and a n +2( p − of ˜ H n ( F j ; Z ( p ) )and ˜ H n +2( p − ( F j ; Z ( p ) ) transgress to the suspensions of Chern class c n +1 and c n + p − modulosome decomposables respectively. DefineΦ : [ S ∧ A i , F j ] → H n ( S ∧ A i ; Z ( p ) ) ⊕ H n +2( p − ( S ∧ A i ; Z ( p ) )by Φ( ξ ) = ξ ∗ ( a n ) ⊕ ξ ∗ ( a n +2( p − ) . Then calculating as in [4] implies the asserted statement. (cid:3)
This theorem can be applied to calculate Samelson products in G . Let Z and Z be spacesand let θ : Z → G and θ : Z → G be maps. Consider the Samelson product h θ , θ i andput Z = Z ∧ Z . Since SU( ∞ ) is homotopy commutative, h θ , θ i lifts to a map ˜ θ : Z → F .Then if Z is a CW-complex consisting only of even dimensional cells, the order of ˜ θ in the coset[ Z, F ] / Im δ ∗ is equal to the order of h θ , θ i . In [6], a convenient lift e θ is constructed such that,if we have the injection Φ of Theorem 4.2, then Φ( e θ ) is calculated from the Chern classes of ρ .We now apply Theorem 4.2 to our case. SHO HASUI, DAISUKE KISHIMOTO, TSELEUNG SO, AND STEPHEN THERIAULT
Proposition 4.3.
For ( G, p, i ) = (E , , , (E , , , there is an injection Φ : [ S ∧ A, F ] → H i +4+2( p − ( S ∧ A ; Z ( p ) ) ⊕ H i +4+4( p − ( S ∧ A ; Z ( p ) ) ∼ = ( Z ( p ) ) satisfying the following, where A = A (2 i + 1 + 2( p − , i + 1 + 4( p − : (1) for ( G, p, i ) = (E , , , Im Φ ◦ δ ∗ is generated by (3 · , − · · and (0 , ) , and alift ˜ θ of h ǫ, λ i can be chosen so that Φ(˜ θ ) = (2 − · , − − · − · ; (2) for ( G, p, i ) = (E , , , is generated by (3 · · · , · · · · and (0 , ) ,and a lift ˜ θ of h ǫ, λ i can be chosen so that Φ(˜ θ ) = (3 · · · , − · · · · .Proof. The Chern classes of ρ are calculated in [6], so we can easily check that the conditionsof Theorem 4.2 are satisfied in our case. We consider the case ( G, p, i ) = (E , , ρ : Σ A → B SU( ∞ ) be the adjoint of ρ ◦ λ i and put ξ = β ∧ ¯ ρ ∈ e K ( S ∧ A ), where β is agenerator of e K ( S ). Let η ∈ e K ( S ∧ A ) be the composite of the pinch map S ∧ A → S anda generator of π ( B U( ∞ )) ∼ = Z (5) . Then by [6], we havech( ξ ) = 58! u + 532 · u , ch( η ) = u . Thus we get Im Φ ◦ δ ∗ as desired. As in [6], we can calculate Φ( e θ ) by using the Chern classesof ρ to get the desired result. The case of ( G, p, i ) = (E , ,
9) is proved similarly. (cid:3)
Corollary 4.4.
We have γ (E ,
5) = 5 and γ (E ,
11) = 11 . The following are proved similarly to Proposition 4.3.
Proposition 4.5.
For ( G, p, i ) = (E , , , (E , , , (E , , , (E , , , (E , , , thereis an injection Φ : [ S ∧ A, F ] → H i +2 p +2 ( S ∧ A ; Z ( p ) ) ∼ = Z ( p ) such that Im Φ ◦ δ ∗ is generated by ∈ Z ( p ) , where A = A (2 i + 1 , i + 1 + 2( p − . Then wehave [ S ∧ A, F ] = 0 . Corollary 4.6. γ (E ,
7) = γ (E ,
11) = γ (E ,
13) = γ (E ,
13) = γ (E ,
19) = 1 . Proposition 4.7.
For ( G, p, i ) = (E , , , there is an injection Φ : [ S ∧ A (35 , , F ] → H ( S ∧ A (35 , Z (13) ) ∼ = Z (13) such that Im Φ ◦ δ ∗ is generated by ∈ Z (13) and a lift ˜ θ of h ǫ, λ i can be chosen such that Φ(˜ θ ) = 2 − · · · · ∈ Z (13) . Corollary 4.8. γ (E ,
13) = 13 . Proposition 4.9.
For ( G, p, i ) = (E , , , there is an injection Φ : π ( F ) → H ( S ; Z (19) ) ∼ = Z (19) DD PRIMARY HOMOTOPY TYPES OF THE GAUGE GROUPS OF EXCEPTIONAL LIE GROUPS 9 such that
Im Φ ◦ δ ∗ is generated by ∈ Z (19) and a lift ˜ θ of h ǫ, λ i i can be chosen such that Φ(˜ θ ) = − · · · · . Corollary 4.10. γ (E ,
19) = 19 . Proposition 4.11. γ (E ,
7) = 7 and γ (E ,
11) = 11 .Proof.
We can prove this proposition in the same manner as above, but it needs more data fromthe Chern characters than that obtained in [6]. Therefore we employ another method also usedin [6]. Consider the case (
G, p ) = (E , B E is given by Z / x , x , x , x , x , x ] for | x i | = i . Let ¯ ǫ : S → B E and ¯ λ : S → B E be the adjointsof ǫ : S → E and λ : S → E , respectively. Since ¯ ǫ is the inclusion of the bottom cell and λ is the inclusion of the direct product factor, we have ¯ ǫ ∗ ( x ) = u and ¯ λ ∗ ( x ) = u for agenerator u i of H i ( S i ; Z / P x has no linear term and includes x x .Suppose that the Samelson product h ǫ, λ i is trivial. Then, taking adjoints, the Whiteheadproduct [¯ ǫ, ¯ λ ] is trivial too, so there is a homotopy commutative diagram S ∨ S
18 ¯ ǫ ∨ ¯ λ / / incl (cid:15) (cid:15) B E S × S µ / / B E . Let u i be a generator of H i ( S i ; Z / µ ∗ ( x ) = u ⊗ µ ∗ ( x ) = u , so0 = P µ ∗ ( x ) = µ ∗ ( P x ) = µ ∗ ( x x ) = u ⊗ u = 0 , a contradiction. Thus the Samelson product h ǫ, λ i is non-trivial. On the other hand, byTheorems 3.1 and 4.1, h ǫ, λ i factors through S ⊂ E and 7 · π ( S ) (11) = 0. Therefore weobtain that γ (E ,
7) = 7 as desired. The case of γ (E ,
11) is similarly verified using the factfrom [6] that P x has no linear term and includes − x x , and the fact that the mod 11cohomology of B E is Z / x , x , x , x , x , x , x ] for | x i | = i . (cid:3) Since F is a retract of E , we can deduce the following from Propositions 4.4 and 4.6. Corollary 4.12.
Possible non-trivial h ǫ, λ i i in F at the prime p are the cases ( p, i ) = (5 , , (7 , and γ (F ,
5) = 5 , γ (F ,
7) = 1 . The case ( G, p ) = (E , , (E , G, p ) = (E , ≃ B (3 , , × B (11 , , × S .As in [15], the H-spaces B (3 , ,
27) and B (11 , ,
35) are direct product factors of the mod p decomposition of SU(18). So we can determine their low dimensional homotopy groups fromthose of SU(18). Therefore Theorems 3.1 and 4.1 imply that h ǫ, λ i is null homotopic. Thuswe get: Proposition 5.1. γ (E ,
7) = 1 . Let A (11 , ,
35) be a homology generating subspace of B (11 , , Proposition 5.2.
For ( G, p ) = (E , , there is an injection Φ : [ S ∧ A (11 , , , F ] → H ( S ∧ A (11 , , Z (7) ) ∼ = Z (7) such that Im Φ ◦ δ ∗ is generated by , implying that [ S ∧ A (11 , , , F ] = 0 .Proof. The proof is similar to that for Proposition 4.3 once we calculate Im Φ ◦ δ ∗ . Let¯ ρ : Σ A (11 , , → B SU( ∞ ) be the adjoint of ρ ◦ λ : A (11 , , → SU( ∞ ) and put ξ = β ∧ ¯ ρ ∈ e K ( S ∧ A (11 , , β ∈ e K ( S ). Then by (4.2) we havech( ξ ) = − u − · u − · u where u i is a generator of H i ( S ∧ A (11 , , Z (7) ) ∼ = Z (7) for i = 14 , ,
38. Thus we getIm Φ ◦ δ ∗ as desired. (cid:3) Corollary 5.3. γ (E ,
7) = 1 . Proposition 5.4. γ (E ,
7) = 7 .Proof.
By Theorems 3.1 and 4.1, the Samelson product h ǫ, λ i in E at p = 7 factors through S ⊂ E , so it is the composite of the pinch map onto the top cell S ∧ A (3 , , → S andan element of π ( S ) (7) . As 7 · π ( S ) (7) = 0, we get γ (E , ≤ h ǫ, λ i is non-trivial. Recall that the mod 7 cohomology of B E is H ∗ ( B E ; Z /
7) = Z / x , x , x , x , x , x , x ] , | x i | = i. If we show that P x includes αx x for α = 0 and does not include the linear term then wemay proceed as in the proof of Proposition 4.11. Let j : Spin(11) → E be the natural inclusion.In [6], it is shown that the generators x i ( i = 4 , , ,
28) are chosen as j ∗ ( x ) = p , j ∗ ( x ) = p + p p , j ∗ ( x ) = p , j ∗ ( x ) = − p p + other terms,where H ∗ ( B Spin(11); Z /
7) = Z / p , p , p , p , p ] and p i is the i -th Pontrjagin class. By themod p Wu formula in [17], we can see that P p = − p p + p p p + other terms , so for degree reasons, we get that P x includes − x x and does not include the linearterm. (cid:3) We next consider the case (
G, p ) = (E , G, p ) = (E , h ǫ, λ i is trivial. Let A be a homology generating subspace of B (23 , , , ⊂ E . DD PRIMARY HOMOTOPY TYPES OF THE GAUGE GROUPS OF EXCEPTIONAL LIE GROUPS 11
Proposition 5.5.
For ( G, p ) = (E , , there is an injection Φ : [ S ∧ A, F ] → H ( S ∧ A ; Z (7) ) ⊕ H ( S ∧ A ; Z (7) ) ∼ = ( Z (7) ) such that Im Φ ◦ δ ∗ is generated by ( − · · · · , · · · · and (0 , ) , and a lift ˜ θ of h ǫ, λ i can be chosen such that Φ(˜ θ ) = ( − · · · · , − · · · · .Proof. Let ξ = β ∧ ¯ ρ ∈ e K ( S ∧ A ) be as in the proof of Proposition 5.2. Let ξ = 12 · ( φ ( ξ ) − ξ ) , ξ = 12 · ·
13 ( φ ( ξ ) − ξ )and let ξ be the composite of the pinch map S ∧ A → S and a generator of π ( B U( ∞ )).Put a = 2 · · , a = 2 · · · , a = 2 · · · · · , a = 2 · · · · ·
61. Thenby [6], we have: ch( ξ ) = a u + a u − a · u + a u ch( ξ ) = 7 a u − · a u + 3 · · · a u ch( ξ ) = − a u + 3 · · · · a u ch( ξ ) = u where u i is a generator of H i ( S ∧ A ; Z (7) ) ∼ = Z (7) for i = 26 , , ,
62. Then Im Φ ◦ δ ∗ can becalculated as asserted, and Φ(˜ θ ) can be calculated as in [6]. (cid:3) Corollary 5.6.
We have γ (E ,
7) = 1 and γ (E ,
7) = 7 .Proof of Theorem 1.1. By Propositions 2.5, 4.11, 5.1, 5.4 and Corollaries 3.5, 4.4, 4.6, 4.8, 4.10,4.12, 5.3, 5.6, we obtain the following table: p ≥ γ (F , p ) 5 γ (E , p ) 5 γ (E , p ) 7 11 1 1 19 1 1 1 1 γ (E , p ) 7
13 1 19 1 1 31 1Therefore γ ( G ) is the product of γ ( G, p ) for p ≥ G = F , E ) and p ≥ G = E , E ), so theassertion of Theorem 1.1 follows from Theorem 2.3. The G case was proved in [12]. (cid:3) References [1] M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces,
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