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Pairings and monomorphisms of classifying spaces
Topology and its Applications 160 (2013) 264–272
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Topology and its Applications www.elsevier.com/locate/topol
Pairings and monomorphisms of classifying spaces Kenshi Ishiguro a,∗ , Shotaro Kudo a , Tomohiro Nakano b a b
Dept. of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan Wajiro junior high school, Higashi-ku, Fukuoka 814-0201, Japan
a r t i c l e
i n f o
Article history: Received 22 May 2012 Received in revised form 18 October 2012 Accepted 18 October 2012 MSC: 55R37 55R35 55P60
a b s t r a c t We consider the maps between classifying spaces of compact Lie groups of the form B K × B L → B G. If the restriction map B L → B G is a weak epimorphism, then the restriction on B K is known to factor through the classifying spaces of the center of the compact Lie group G. Suppose H is a semi-simple subgroup of a connected compact Lie group G with rank( H ) = rank(G ). Replacing the weak epimorphism B L → B G by the map B H → B G, analogous results are obtained. We also consider some monomorphisms of classifying spaces of compact Lie groups, such as BSO(n) → BSU (n). Our proof will make use of admissible maps. © 2012 Elsevier B.V. All rights reserved.
Keywords: Classifying space Lie group p-Compact group p-Completion K-theory Admissible map
We consider the pairing problem of classifying spaces of compact Lie groups for monomorphisms. Recall in general that for a map f : Y → Z , the set of the homotopy classes of axes, denoted by f ⊥ ( X , Z ), consists of all homotopy classes of maps α : X → Z such that there is a map (called a pairing) μ : X × Y → Z with restrictions (axes) μ| X α and μ|Y f . In the case of classifying spaces [10], we have the following commutative diagram:
BL f
BK × BL
μ
BG
α
BK Here we denote α ∈ f ⊥ ( B K , B G ). Let f = Bi where i : L → G is a monomorphism of Lie groups. We will obtain a sufficient condition for ( Bi )⊥ ( B K , B G ) = 0 and calculate some pairings. The first author has studied the pairing problem of classifying spaces for weak epimorphisms. By Theorem 1 of [10], we see that if the restriction map B L → B G is a weak epimorphism, then the restriction on B K factors through the classifying spaces of the center of the compact Lie group G. A generalization for p-compact groups can be found in [11]. A p-compact
*
Corresponding author. E-mail addresses: [email protected] (K. Ishiguro), [email protected] (S. Kudo).
0166-8641/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2012.10.018
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265
group [6] and [17], is a loop space X such that X is F p -finite and that its classifying space B X is F p -complete. The pcompletion of a compact Lie group G is a p-compact group if π0 G is a p-group. It is worth to recall here a group theoretical analog. Suppose ρ : L → G and α : K → G are homomorphisms. If there is a pairing homomorphism μ : K × L → G with μ| K = α and μ|L = ρ , then the image α ( K ) must be contained in the centralizer of ρ ( L ) in G, denoted by C G (ρ ). Theorem 1. Suppose that K is a compact Lie group, and that a connected compact Lie group H is a semi-simple subgroup of a connected compact Lie group G with rank( H ) = rank(G ). Let i : H → G be the inclusion. If α ∈ ( Bi )⊥ ( B K , B G ), then the following hold: (1) The map α : B K → B G factors through B π0 K up to homotopy under the map induced by the projection q : K → π0 K . In particular, if K is connected, the map α is null homotopic. (2) There is a homomorphism ρ : π0 K → G such that α B ρ ◦ Bq, and the image of the homomorphism ρ (π0 K ) is contained in the centralizer C G ( H ). Theorem 2. For the inclusions i : SU (m) → SU (n) and j : Sp(m) → Sp(n) with m n, we have the following: (1) ( Bi )⊥ (BSU (k), BSU (n)) = [BSU (k), BSU (n − m)]. (2) ( B j )⊥ (BSp(k), BSp(n)) = [BSp(k), BSp(n − m)]. A result of [12] shows that for the inclusion i : SU (n) → U (n), if a connected compact Lie group K is semi-simple, then ( Bi )⊥ ( B K , BU (n)) = 0. Similar results for other classical Lie groups are obtained in [15]. Theorem 3. For the inclusion i : SO(n) → SU (n) with n 3, if G is a connected compact Lie group, then any map in ( Bi )⊥ ( B G , BSU (n)) is null homotopic:
( Bi )⊥ B G , BSU (n) = 0. We will prove the p-completed version of this result. So Theorem 3 is its easy consequence. Some of the results in this paper first appeared in the third author’s master thesis. 1. Ranks of pairing maps We will prove Theorem 1 in this section. To do so, we need a few basic results, mostly something about ranks of admissible maps. Lemma 1.1. Let G be semi-simple. For any infinite order element α of a maximal torus T G , there is an element λ contained in the normalizer N T G such that λα λ−1 α −1 is not contained in the center Z (G ). Proof. Since G is semi-simple, the order of Z (G ) is finite. Let | Z (G )| = n. Now suppose λα λ−1 α −1 ∈ Z (G ) for any λ ∈ N T G . Then we would see λα λ−1 α −1 = ζ for some ζ ∈ Z (G ), and hence λαn λ−1 = (λα λ−1 )n = ζ n αn = αn . This means that αn is fixed by the action of the Weyl group W (G ) = N T G / T G . Consequently the infinite set {(αn )k | k ∈ Z} could be contained W (G ) W (G ) . This is, however, a finite set, since T G / Z (G ) is an elementary abelian 2-group [7, Remark 1.5]. This in the set T G contradiction completes the proof. 2 ∧ Here we first recall the kernel of a map f : B L → ( B G )∧ p [9,5], and [16], where X p denotes the p-completion of a space X . Let T (or T L ) be a maximal torus of the Lie group L. Suppose N p T denotes the inverse image in the normalizer of a maximal torus T of a p-Sylow subgroup W p of the Weyl group of L. We define a subgroup N p ∞ T of N p T as follows:
T
Z/ p ∞
NpT
Wp
N p∞ T
Wp
Here Z/ p ∞ ⊂ T is the subgroup of elements whose order is a power of p. The kernel of a map f : B L → ( B G )∧p is defined in [9, §1] as follows:
Ker f = {x ∈ N p ∞ T | f | B x 0}. Here x denotes the subgroup of N p ∞ T generated by x. We note that Ker f is a group. Next we recall subgroups related to the center of connected compact Lie groups, [9]. Let Z ( L ) denote the center of L and let W ( L ) denote its Weyl group. If L is a simply-connected simple Lie group other than the exceptional Lie group G 2 , we define
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Z (L, p) = If L = G 2 , then
Z (L) a maximal torus of L
Z (G 2 , p ) =
if | W ( L )| ≡ 0 mod p , if | W ( L )| ≡ 0 mod p .
Z (G 2 ) the center of the subgroup SU (3) of G 2 a maximal torus of G 2
if p = 2, if p = 3, if p 5.
According to [9, Lemma 2], if a map f : B L → ( B G )∧ p is not null homotopic, then Ker f is included in Z ( L , p ). Lemma 1.2. Let K , L and G be connected compact Lie groups. Suppose L is simple. For a map f : B K × B L → B G, if there is a subgroup S 1 of K × L, but S 1 ⊂ K , such that f | B S 1 0, then f | B L 0. Proof. Since S 1 ⊂ K , we can find (x, y ) ∈ S 1 ∩ Ker( f p∧ ) ⊂ K × L where y is an infinite order element in L. Lemma 1.1
/ Z ( L ). Since (x, y ) ∈ Ker( f p∧ ), we see that (x−1 , y −1 ), (x, λ y λ−1 ) ∈ Ker f . says that there is λ ∈ N T L such that λ y λ−1 y −1 ∈ Consequently we obtain the following:
x, λ y λ−1 · x−1 , y −1 = 1, λ y λ−1 y −1 ∈ Ker f p∧ .
Since λ y λ−1 y −1 ∈ / Z ( L ) and L is simple, there is a prime p such that | W ( L )| ≡ 0 mod p, and the composite map B L → B K × B L → B G → ( B G )∧ p is null homotopic. Therefore f | B L 0 by [9]. 2 Let G and G be connected compact Lie groups with maximal tori T and T , respectively. For a map f : B G → B G, we can find a homomorphism (admissible map), φ : T → T by [2, Theorem 1.1] which makes the following diagram commutative:
H ∗ ( B G ; Q)
H ∗ ( B T ; Q)
f∗
( B φ)∗
H ∗ ( B G ; Q)
H ∗ ( B T ; Q)
Here the vertical maps are injective. It is well known that the rational cohomology H ∗ ( B G , Q) is isomorphic to the ring of invariants H ∗ ( B T ; Q) W (G ) . We define the rank of f : B G → B G, denoted by rank( f ), as the rank of the linear map ( B φ)∗ : H 2 ( B T ; Q) → H 2 ( B T ; Q). This is well defined, since the choice of φ is unique up to conjugation. For a finite covering G → G we see H ∗ ( B G ; Q) ∼ = H ∗ ( B G ; Q). Thus, using a universal finite covering, we may assume that the connected compact Lie groups are products of simply-connected simple Lie groups and a torus when we discuss the rank of f . Theorem 1.1. Let K , L and G be connected compact Lie groups. If the following hold:
μ : B K × B L → B G is a map with μ| B K = α and μ| B L = f , then
(1) rank(μ) rank( f ) + rank(α ). (2) If L is semi-simple, then rank(μ) = rank( f ) + rank(α ). Proof. Recall that if n = rank(G ), then H ∗ ( B T G ; Q) is a polynomial ring generated by n variables of degree 2. So H 2 ( B T G ; Q) is regarded as an n-dimensional vector space over Q. For the map μ, we have the following commutative diagram:
H ∗ ( B G ; Q)
H ∗ ( B T G ; Q)
μ∗
( B φ)∗
H ∗ ( B K × B L ; Q)
H ∗ ( B T K × B T L ; Q)
Let k = rank( K ) and = rank( L ). We notice that the admissible map ( B φ)∗ : H 2 ( B T G ; Q) → H 2 ( B T K × B T L ; Q) is presented by a (k + ) × n matrix. This matrix is decomposed as a k × n matrix and an × n matrix obtained from α ∗ and f ∗ respectively. Consequently rank(μ) rank( f ) + rank(α ). It remains to show rank(μ) rank( f ) + rank(α ), if L is semi-simple. The universal covering L of L is a product group of 1-connected simple Lie groups. Since H ∗ ( B L ; Q) ∼ L ; Q), as far as the rank of a map is concerned, we may assume = H ∗ ( B m L = i =1 L i where each L i is simple. We notice that rank( f | BL i ) is equal to either 0 or rank( L i ). We see that rank( f ) =
rank( f | BL i ). Let J = {i | rank( f | BL i ) = 0} and let L = respectively, such that i
rank( B φ| B T 1 ) = rank( T 1 ) = rank(α ),
j∈ J
L j . We can find subtori T 1 and T 2 (= T L ) of T K and T L
rank( B φ| B T 2 ) = rank( T 2 ) = rank( f ).
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If rank(μ) rank( f ) + rank(α ), then there is a subgroup S 1 of T 1 × T 2 , but S 1 ⊂ T 1 , such that B φ| B S 1 0. Lemma 1.2 implies that f | BL j 0 for some L j . Since rank( f | BL j ) = 0, this contradiction completes the proof. 2 Proof of Theorem 1. First we show the part (1). Let K 0 denote the connected component of the compact Lie group K . Suppose μ : B K 0 × B H → B G is the restriction of a pairing B K × B H → B G with μ| B K 0 = α | B K 0 and μ| B H = Bi. Since H is semi-simple with rank( H ) = rank(G ) and rank(μ) rank(G ), we see that Theorem 1.1 implies rank(α | B K 0 ) = 0. Therefore the map α | B K 0 is null homotopic, and α factors through B π0 K . Next we show the part (2). Let α : B π0 K → B G be the induced map. Applying [10, Corollary 1.2] to a restriction of the pairing B π0 K × B T H → B G, we see that the map α factors through BC G ( T H )0 at any prime p. Since G is connected, we see C G ( T H ) = T H by [1, Proposition 4.25]. Let π = π0 K and T = T H . By Proposition 4 of [13] we know that, since T is abelian, any map B π → B T is induced by a homomorphism. Since π is a finite group, we notice [14, §2] that [ B π , B T ] = ∧ ∧ p [ B π p , B T p ]. Therefore α is induced by a homomorphism. 2 Remark 1.1. If the subgroup H of G in Theorem 1 is not semi-simple, the result does not hold. A counterexample is given as follows: Take a maximal torus T G of G so that rank( T G ) = rank(G ). The multiplication map B T G × B T G → B T G composite with Bi : B T G → B G gives a pairing B T G × B T G → B G. We notice that Bi ∈ ( Bi )⊥ ( B T G , B G ). The monomorphism Bi, however, does not factor through the one-point space B π0 T G . Remark 1.2. The p-completion of a connected compact Lie group is a p-compact group, [6]. Suppose f : X → Y is a monomorphism of p-compact groups. If rank( X ) = rank(Y ) and | W ( X )| = | W (Y )|, then f is an isomorphism. Let i : H → G be as in Theorem 1. Furthermore, the orders of their Weyl groups are the same. Then one can show that H is isomorphic to G. We add that, according to [4, Lemma 4.6], the map f is an isomorphism if and only if the mod p cohomological dimensions of X and Y are the same. Theorem 1.2. Let H be a connected subgroup of a connected compact Lie group G. For the inclusion i : H → G, if K is simple and rank(G ) < rank( K ) + rank( H ), then ( Bi )⊥ ( B K , B G ) = 0. Proof. For
α ∈ ( Bi )⊥ ( B K p∧ , B G ∧p ), let μ be the pairing B K p∧ × B H ∧p → B G ∧p , and let φ be its admissible map: μ
B K p∧ × B H ∧ p
( B T K )∧p × ( B T H )∧p
Bφ
BG∧ p
( B T G )∧p
Since rank(G ) < rank( K ) + rank( H ), then rank(φ) < rank( K ) + rank( H ). Notice that φ| T H is one-to-one. Consequently we can find a subgroup S 1 of K × H , but S 1 ⊂ H , such that μ| B S 1 0. Since K is simple, Lemma 1.2 shows μ| B K 0 and therefore ( Bi )⊥ ( B K , B G ) = 0. 2 2. Classical Lie groups and admissible maps In this section we will consider the pairing problem of classical Lie groups. We prove Theorem 2 and the p-completed version of Theorem 3. Proof of Theorem 2. Suppose α ∈ ( Bi )⊥ (BSU (k), BSU (n)). The pairing μ : BSU (k) × BSU (m) → BSU (n) for this map α is also a pairing for Bi. So we see Bi ∈ (α )⊥ (BSU (m), BSU (n)). Hence, for any subgroup Z/ p of SU (m), it follows that Bi | B Z/ p ∈ (α )⊥ ( B Z/ p , BSU (n)∧p ). The map α : BSU(k)∧p → BSU (n)∧p factors through BC SU(n) (Z/ p )∧p up to homotopy [10, Corollary 1.2]. Choose a prime number p which divides m, and let ζ be a primitive p-th root of unity. We define a scalar matrix [ζ ] = diag(ζ, ζ, . . . , ζ ) ∈ SU (m). We will compute the centralizer C SU(n) (Z/ p ):
and
[ζ ] 0 0
I
A C
B D
A · C
B D
=
[ζ ] A [ζ ] B C
D
A [ζ ] B [ζ ] 0 · = . 0 I C [ζ ] D
Consequently B = 0 and C = 0. Thus C SU(n) (Z/ p ) is a subgroup of U (m) × U (n − m). By Theorem 1 of [12], we see that the pairing μ maps BSU (k) trivially into BU (m). Any map from BSU (k) to BU (n − m) factors through BSU (n − m), since BSU (k) is 3-connected. This implies the desired result. An analogous argument is applicable for the case of symplectic groups. 2
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We recall some arguments of admissible maps for the mod p-cohomology and then for the p-adic K-theory. For connected compact Lie groups G and K with maximal tori T G and T K respectively, suppose H ∗ ( B G ; F p ) ∼ = H ∗ ( B T G ; F p ) W (G ) and H ∗ ( B K ; F p ) ∼ = H ∗ ( B T K ; F p )W ( K ) . We note that H ∗ ( B G ; F p ) is isomorphic to H ∗ ( B T G ; F p )W (G ) , for instance, if p does not divide the order of W (G ). For any map f : B G → B K we have the commutative diagram: φf
H ∗(B T K ; Fp )
H ∗(B K ; Fp )
H ∗(B T G ; Fp )
H ∗(B G ; Fp )
f∗
Here φ = φ f is admissible [3] and [2]; namely for any w ∈ W (G ) we can find w ∈ W ( K ) such that w φ = φ w . Recall that H ∗ ( B T n ; F p ) = F p [t 1 , t 2 , . . . , tn ] is a polynomial ring in n variables of degree 2. Hence the admissible map φ over the Steenrod algebra can be regarded as a rank(G ) × rank( K ) matrix. Next consider, [8] and [18], the admissibility for K-theory with coefficients in Z∧ p , the ring of p-adic integers. Recall that
∧ ∧ ∧ W (G ) . Here note K ( B T n ; Z∧ p ) = Z p [[x1 , x2 , . . . , xn ]], the ring of power series, and that K ( B G ; Z p ) is isomorphic to K ( B T G ; Z p ) ∞ that each xi + 1 is the pullback of the Hoph line bundle over C P via the i-th projection map. For any map f : B G → B K we have the commutative diagram:
φ
K ( B T K ; Z∧ p)
K ( B K ; Z∧ p)
K ( B T G ; Z∧ p)
f∗
K ( B G ; Z∧ p)
Again, [8, Lemma 2.1], for any w ∈ W (G ) we can find w ∈ W ( K ) such that w φ = φ w . We will use the following basic results about the maximal tori and the Weyl groups for SO(n) and SU (n) , in order to
cos θ − sin θ prove Theorem 2.1, the p-completed version of Theorem 3. We note first that SO(2) consists of the matrices , sin θ cos θ
and that
T m = SO(2) × · · · × SO(2) ⊂ SO(2m) ⊂ SO(2m + 1) is a maximal torus in both SO(2m) and SO(2m + 1). Let T be the set of diagonal matrices
⎛
⎞
e i θ1
⎜ ⎜ ⎝
e
⎟ ⎟ ⎠
i θ2
..
. e i θn
θ − sin θ ≡ 0 mod 2π . Then T is a maximal torus of SU (n). The 2 × 2 matrix A = cos is diagonalizable. Let sin θ cos θ iθ
1 i 1 e 0 − 1 . This induces a monomorphism j : SO(n) → SU (n) by j ( B ) = Q B Q −1 where P=√ ∈ U (2). Then P A P = 2 1 −i 0 e −i θ B ∈ SO(n) and
where
n
k=1 θk
⎛
⎞
P
⎜
Q =⎜ ⎝
⎟ ⎟. ⎠
P
..
. P
Since Bi B j for the inclusion i : SO(n) → SU (n), we see that, for n = 2m, the admissible map is given by the following m × (2m − 1) matrix:
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
1
−1
⎞ 1 −1
..
⎟ ⎟ ⎟. ⎟ ⎠
.
1
−1 1
m ∧ W (SO(2m)) , where W (SO(2m)) is generated by the permutation representation ∼ We note that K (BSO(2m); Z∧ p ) = K (B T ; Zp )
of Σm and the following m × m matrix
τ:
K. Ishiguro et al. / Topology and its Applications 160 (2013) 264–272
⎛ ⎜ ⎜ τ =⎜ ⎜ ⎝
269
⎞
−1
⎟ ⎟ ⎟. ⎟ ⎠
−1 1
..
. 1
Namely, the Weyl group W (SO(2m)) permutes the xi ’s together with an even number of changes of signs for ∧ K ( B T m ; Z∧ p ) = Z p [[x1 , x2 , . . . , xm ]].
m ∧ W (SO(2m+1)) , where the Weyl group W (SO(2m + 1)) is ∼ If n = 2m + 1, we see that K (BSO(2m + 1); Z∧ p ) = K (B T ; Zp ) generated by the permutation representation of Σm and the following m × m matrix σ :
⎛ ⎜
⎞
−1
⎟ ⎟. ⎠
1
σ =⎜ ⎝
..
. 1
Namely W (SO(2m + 1)) permutes the xi ’s together with an arbitrary number of changes of signs. Finally recall that W (SU (n)) is isomorphic to the symmetric group Σn . The representation as a subgroup of G L (n − 1, Z∧ p)
n−1 ∼ which makes K (BSU (n); Z∧ ; Z∧p )W (SU(n)) is generated by the permutation representation of Σn−1 together with p ) = K (B T the following (n − 1) × (n − 1) matrix:
⎛
⎞ −1 . .. ⎜ .. ⎟ . ⎜ ⎟. ⎝0 ⎠ 1 −1 0 · · · 0 −1 1
0
n
= 0, then the representation of W (SU (n)) can be regarded
0 1as the permutation
1 0 0 repre0 sentation of Σn . For instance, when n = 4, the transposition (1, 2) corresponds to the matrix 1 0 0 , (2, 3) to 0 0 1 , and 001 010
1 0 −1 (3, 4) to 0 1 −1 , respectively. We will use this convention from now on. In other words, if we let
i =1 xi
0 0 −1
To prove Theorem 3, it suffices to prove the following p-completed version, since for a map p-completion α ∧ p is null homotopic, so is the map α .
α : B G → BSU(n), if the
Theorem 2.1. For the inclusion i : SO(n) → SU (n) with n 3, if f = ( Bi )∧ p and G is a connected compact Lie group, then any map in ∧ f ⊥(B G∧ p , BSU (n) p ) is null homotopic for any prime p.
Proof. First we consider the case n = 2m. For commutative diagram:
α ∈ f ⊥ ( B G ∧p , BSU(2m)∧p ) there is a pairing map μ which gives the following
BSO(2m)∧ p f
μ
∧ BG∧ p × BSO(2m) p
BSU (2m)∧ p
α
BG∧ p
∧ ∧ ∼ k ∧ W (G ) . Consider the admissible map φ which gives Assume rank(G ) = k so that K ( B G ∧ p ; Zp ) = K (B G ; Zp ) = K (B T ; Zp ) the commutative diagram:
K ( B T 2m−1 ; Z∧ p)
K (BSU (2m); Z∧ p)
φ
μ∗
m ∧ K ( B T k ; Z∧ p ) ⊗ K (B T ; Zp )
∧ K ( B G ; Z∧ p ) ⊗ K (BSO(2m); Z p )
We recall that φ can be regarded as a (k + m) × (2m − 1) matrix. If φα is a k × (2m − 1) matrix expressing the admissible map which covers α ∗ and φ f is an m × (2m − 1) matrix expressing the admissible map which covers f ∗ , then the (k + m) × (2m − 1) matrix φ is decomposed as follows:
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φ=
φα φf
.
Recall here the following:
⎛
⎞
−1
1
⎜ ⎜ φf = ⎜ ⎜ ⎝
1 −1
..
⎟ ⎟ ⎟. ⎟ ⎠
. −1
1
1 For any w ∈ W (SO(2m))
Ik 0
0 w
φα φf
we can find w
=
φα φf
∈ W (SU (2m)) such that
· w
where I k denotes the k × k identity matrix. In fact, we will see that, for w ∈ W (SO(2m)), the matrix w ∈ W (SU (2m)) is uniquely determined. Notice here that the column vectors of a matrix expressing an element of W (SU (2m)) consists of 2m − 1 vectors out of the set of 2m vectors {e 1 , e 2 , . . . , e 2m−1 , v } where {e 1 , e 2 , . . . , e 2m−1 } is the standard basis and v = t (−1 − 1 · · · − 1). Using this result and our convention, if w = (i , i + 1), one can show w = (2i − 1, 2i + 1)(2i , 2i + 2). And if w = τ , then w = (1, 2)(3, 4). We will show φα = 0. Let φα is expressed as follows:
⎛a
11
a12
φα = ⎝ ... ak1
.. . ak2
· · · a12m−1 ⎞ .. ⎠ . . · · · ak2m−1
Since φα w = φα for any w ∈ W (SU (2m)), we see that ai j = ai for all 1 j , 2m − 1, and ai1 = −(2m − 1)ai1 so that 2mai1 = 0. Since the p-adic integers Z∧ p is torsion-free, each a i1 must be zero, and hence φα = 0. Consequently, K (μ) = 0. A result of Zabrodsky, [19, Theorem 1], shows that ( Bi )⊥ ( B G , BSU (2m)) = 0. It remains to consider the case n = 2m + 1.
BSO(2m + 1)∧ p f
μ
∧ BG∧ p × BSO(2m + 1) p
BSU (2m + 1)∧ p
α
BG∧ p
Again, for w ∈ W (SO(2m + 1)), one can show that the matrix w ∈ W (SU (2m + 1)) is uniquely determined. If w = (i , i + 1), then w = (2i − 1, 2i + 1)(2i , 2i + 2). And if w = σ , then w = (1, 2). Similarly, we can show φα = 0, and hence ( Bi )⊥ ( B G , BSU (2m + 1)) = 0. 2 Remark 2.1. The Lie group SU (n) is a subgroup of the symplectic group Sp(n). Here we consider what if SU (n) is replaced by Sp(n) in Theorem 3. It seems that the things are different. First, in the proof of the theorem, a property of admissible maps has been important. Namely, for any w ∈ W (SO(n)), we see that w ∈ W (SU (n)) is uniquely determined. In the case of the inclusion j : SO(n) → Sp(n), this property no longer holds. For instance, if f = ( B j )∧ p , then φ f = (1, −1, 0). We recall
that W (SO(3)) = {±1} and W (Sp(3)) = Σ3 ,
−1 0 0 0 10 0 01
. Take w = −1. Then w can be either − I 3 or
0 1 0 100 001
. Moreover, for
− I 2 , four matrices can be w . Apparently our proof for SU(n) is not applicable for Sp(n). In fact we will see
n = 4, if w = ( B j )⊥ ( B S 1 , BSp(n)) = 0. To see this, consider the homomorphisms of Lie groups:
SO(n) × S 1
U (n) × Z (U (n))
U (n)
Sp(n)
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where the map SO(n) → U (n) is the canonical inclusion and S 1 → Z (U (n)) is the identity map. We can see that the axis B S 1 → BSp(n) of the induced map BSO(n) × B S 1 → BSp(n) is an essential map. Finally we consider the mod p-cohomology version. The result is slightly different from the K-theory case. Theorem 2.2. Let i : SO(n) → SU (n) be the inclusion for n 3. Suppose that a space X is a connected p-compact group with maximal torus T X and Weyl group W ( X ) such that the mod p cohomology H ∗ ( B X ; F p ) is isomorphic to the ring of invariants H ∗ ( B T X ; F p ) W ( X ) . Suppose the following diagram is commutative over the Steenrod algebra A p :
H ∗ (BSO(n); F p ) ( Bi )∗ ψ
H ∗ (BSU (n); F p )
H ∗ ( B X ; F p ) ⊗ H ∗ (BSO(n); F p ) β
H ∗(B X ; Fp ) Then rank(φβ ) 1. Moreover, β = 0 if p does not divide n. Proof. The argument here is similar to the one used in the proof of Theorem 3. First we treat the case that a prime p is odd and n = 2m. Assume rank( X ) = k so that H ∗ ( B X ; F p ) ∼ = H ∗ ( B T k ; F p )W ( X ) . We notice that H ∗ (BSO(2m); F p ) ∼ = H ∗ ( B T m ; F p )W (SO(2m)) and H ∗ (BSU (2m); F p ) ∼ = H ∗ ( B T 2m−1 ; F p )W (SU(2m)) , since p is odd. Consider the admissible map φ which gives the following commutative diagram:
H ∗ ( B T 2m−1 ; F p )
φ
H ∗ (BSU (2m); F p )
ψ
H ∗(B T k ; Fp ) ⊗ H ∗(B T m; Fp )
H ∗ ( B X ; F p ) ⊗ H ∗ (BSO(2m); F p )
We note here that the (k + m) × (2m − 1) matrix φ is decomposed as follows:
φ=
φβ
.
φ( Bi )∗
As a subgroup of G L (2m, F p ), the Weyl group W (SO(2m)) is generated by the permutation representation of Σm and the following m × m matrix:
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
−1
⎞
⎟ ⎟ ⎟. ⎟ ⎠
−1 1
..
. 1
For φβ = (ai j ), as seen in the proof of Theorem 3, it follows that ai j = ai for all 1 j , 2m − 1, and ai1 = −(2m − 1)ai1 . Thus we see that rank(φβ ) 1. Notice that 2mai1 = nai1 = 0. If p does not divide n, then each ai1 is zero. Therefore φβ = 0, and hence β = 0. The case of n = 2m + 1 is similarly proved. Suppose next p = 2. Doubling the degree, we see that the A2 -algebra structure of H ∗ (BSO(n); F2 ) can be same as that of H ∗ (BSU (n); F2 ). Namely, if we express H ∗ (BSO(n); F2 ) = F2 [x2 , x3 , . . . , xn ], then H ∗ (BSU (n); F2 ) is isomorphic to F2 [x22 , x23 , . . . , xn2 ]. For n 3, notice that H ∗ (BSU (n); F2 ) ∼ = H ∗ ( B T n−1 ; F2 )W (SU(n)) . Consider the admissible map φ which gives the commutative diagram:
H ∗ ( B T n −1 ; F p )
H ∗ (BSU (n); F p )
φ
H ∗ ( B T k ; F p ) ⊗ H ∗ ( B T n −1 ; F p )
ψ
Im ψ
The (k + m) × (n − 1) matrix φ is decomposed as follows:
φ=
φβ I n −1
.
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Since φα w = φα for any w ∈ W (SU (n)), we see that rank(φβ ) 1. We see also that if n is odd, then β = 0. This completes the proof. 2 Remark 2.2. In the case n ≡ 0 mod p, there is an A p -map
H ∗ BSU (n); F p → H ∗ ( B X ; F p ) ⊗ H ∗ BSO(n); F p
∗ ∗ for B X = ( B S 1 )∧ p such that the restriction H (BSU (n); F p ) → H ( B X ; F p ) is non-zero. Let ζ ∈ C be a primitive p-th root of unity so that ζ n = 1. A homomorphism ϕ : SO(n) × Z/ p → SU (n) is induced by ϕ ( A , ζ ) = A · (ζ I n ), and is delooped to the following map:
B ϕ : BSO(n) × B Z/ p → BSU (n). This gives an A p -map
H ∗ BSU (n); F p → H ∗ ( B Z/ p ; F p ) ⊗ H ∗ BSO(n); F p
which factors through H ∗ ( B S 1 ; F p ) ⊗ H ∗ (BSO(n); F p ) for dimensional reason. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
J.F. Adams, Lectures on Lie Groups, W.A. Benjamin, Inc., New York–Amsterdam, 1969. J.F. Adams, Z. Mahmud, Maps between classifying spaces, Inventiones Math. 35 (1976) 1–41. J.F. Adams, C.W. Wilkerson, Finite H-spaces and algebras over the Steenrod algebra, Ann. of Math. 111 (1980) 95–143. N. Castellana, R. Levi, D. Notbohm, Homology decompositions for p-compact groups, Adv. Math. 216 (2) (2007) 491–534. W.G. Dwyer, C.W. Wilkerson, Mapping spaces of nullhomotopic maps, Astérisque 191 (1990) 97–108. W.G. Dwyer, C.W. Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. 139 (2) (1994) 395–442. W.G. Dwyer, C.W. Wilkerson, Centers and Coxeter elements, in: Contemp. Math., vol. 271, Amer. Math. Soc., Providence, RI, 2001, pp. 53–75. K. Ishiguro, Classifying spaces and p-local irreducibility, J. Pure Appl. Algebra 49 (3) (1987) 253–258. K. Ishiguro, Retracts of classifying spaces, in: Adams Memorial Symposium on Algebraic Topology, vol. 1, in: London Math. Soc. Lecture Notes, vol. 175, Cambridge Univ. Press, 1992, pp. 271–280. K. Ishiguro, Classifying spaces and homotopy sets of axes of pairings, Proc. Amer. Math. Soc. 124 (12) (1996) 3897–3903. K. Ishiguro, Pairings of p-compact groups and H -structures on the classifying spaces of finite loop spaces, Publ. Res. Inst. Math. Sci. 34 (6) (1998) 567–578. K. Ishiguro, M. Tokuzawa, Unitary groups and pairings of classifying spaces, Hiroshima Math. J. 29 (2) (1999) 245–253. R. Lashof, P. May, G. Segal, Equivariant bundles with abelian structural group, in: Contemp. Math., vol. 19, AMS, 1983, pp. 167–176. G. Mislin, C. Thomas, On the homotopy set [ B π , B G ] with π finite and G a compact connected Lie group, Quart. J. Math. Oxford Ser. (2) 40 (157) (1989) 65–78. T. Nakano, Pairing problem of classifying spaces for compact Lie groups, Master thesis, Fukuoka University, 2009 (in Japanese). D. Notbohm, Kernels of maps between classifying spaces, Israel J. Math. 87 (1994) 243–256. D. Notbohm, Classifying spaces of compact Lie groups and finite loop spaces, in: Handbook of Algebraic Topology, North-Holland, 1995, pp. 1049–1094. C.W. Wilkerson, Lambda-rings, binomial domains, and vector bundles over CP(∞), Comm. Algebra 10 (3) (1982) 311–328. A. Zabrodsky, Maps between classifying spaces, in: Ann. of Math. Stud., vol. 113, Princeton Univ. Press, 1987, pp. 228–246.
Contents lists available at SciVerse ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Pairings and monomorphisms of classifying spaces Kenshi Ishiguro a,∗ , Shotaro Kudo a , Tomohiro Nakano b a b
Dept. of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan Wajiro junior high school, Higashi-ku, Fukuoka 814-0201, Japan
a r t i c l e
i n f o
Article history: Received 22 May 2012 Received in revised form 18 October 2012 Accepted 18 October 2012 MSC: 55R37 55R35 55P60
a b s t r a c t We consider the maps between classifying spaces of compact Lie groups of the form B K × B L → B G. If the restriction map B L → B G is a weak epimorphism, then the restriction on B K is known to factor through the classifying spaces of the center of the compact Lie group G. Suppose H is a semi-simple subgroup of a connected compact Lie group G with rank( H ) = rank(G ). Replacing the weak epimorphism B L → B G by the map B H → B G, analogous results are obtained. We also consider some monomorphisms of classifying spaces of compact Lie groups, such as BSO(n) → BSU (n). Our proof will make use of admissible maps. © 2012 Elsevier B.V. All rights reserved.
Keywords: Classifying space Lie group p-Compact group p-Completion K-theory Admissible map
We consider the pairing problem of classifying spaces of compact Lie groups for monomorphisms. Recall in general that for a map f : Y → Z , the set of the homotopy classes of axes, denoted by f ⊥ ( X , Z ), consists of all homotopy classes of maps α : X → Z such that there is a map (called a pairing) μ : X × Y → Z with restrictions (axes) μ| X α and μ|Y f . In the case of classifying spaces [10], we have the following commutative diagram:
BL f
BK × BL
μ
BG
α
BK Here we denote α ∈ f ⊥ ( B K , B G ). Let f = Bi where i : L → G is a monomorphism of Lie groups. We will obtain a sufficient condition for ( Bi )⊥ ( B K , B G ) = 0 and calculate some pairings. The first author has studied the pairing problem of classifying spaces for weak epimorphisms. By Theorem 1 of [10], we see that if the restriction map B L → B G is a weak epimorphism, then the restriction on B K factors through the classifying spaces of the center of the compact Lie group G. A generalization for p-compact groups can be found in [11]. A p-compact
*
Corresponding author. E-mail addresses: [email protected] (K. Ishiguro), [email protected] (S. Kudo).
0166-8641/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2012.10.018
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group [6] and [17], is a loop space X such that X is F p -finite and that its classifying space B X is F p -complete. The pcompletion of a compact Lie group G is a p-compact group if π0 G is a p-group. It is worth to recall here a group theoretical analog. Suppose ρ : L → G and α : K → G are homomorphisms. If there is a pairing homomorphism μ : K × L → G with μ| K = α and μ|L = ρ , then the image α ( K ) must be contained in the centralizer of ρ ( L ) in G, denoted by C G (ρ ). Theorem 1. Suppose that K is a compact Lie group, and that a connected compact Lie group H is a semi-simple subgroup of a connected compact Lie group G with rank( H ) = rank(G ). Let i : H → G be the inclusion. If α ∈ ( Bi )⊥ ( B K , B G ), then the following hold: (1) The map α : B K → B G factors through B π0 K up to homotopy under the map induced by the projection q : K → π0 K . In particular, if K is connected, the map α is null homotopic. (2) There is a homomorphism ρ : π0 K → G such that α B ρ ◦ Bq, and the image of the homomorphism ρ (π0 K ) is contained in the centralizer C G ( H ). Theorem 2. For the inclusions i : SU (m) → SU (n) and j : Sp(m) → Sp(n) with m n, we have the following: (1) ( Bi )⊥ (BSU (k), BSU (n)) = [BSU (k), BSU (n − m)]. (2) ( B j )⊥ (BSp(k), BSp(n)) = [BSp(k), BSp(n − m)]. A result of [12] shows that for the inclusion i : SU (n) → U (n), if a connected compact Lie group K is semi-simple, then ( Bi )⊥ ( B K , BU (n)) = 0. Similar results for other classical Lie groups are obtained in [15]. Theorem 3. For the inclusion i : SO(n) → SU (n) with n 3, if G is a connected compact Lie group, then any map in ( Bi )⊥ ( B G , BSU (n)) is null homotopic:
( Bi )⊥ B G , BSU (n) = 0. We will prove the p-completed version of this result. So Theorem 3 is its easy consequence. Some of the results in this paper first appeared in the third author’s master thesis. 1. Ranks of pairing maps We will prove Theorem 1 in this section. To do so, we need a few basic results, mostly something about ranks of admissible maps. Lemma 1.1. Let G be semi-simple. For any infinite order element α of a maximal torus T G , there is an element λ contained in the normalizer N T G such that λα λ−1 α −1 is not contained in the center Z (G ). Proof. Since G is semi-simple, the order of Z (G ) is finite. Let | Z (G )| = n. Now suppose λα λ−1 α −1 ∈ Z (G ) for any λ ∈ N T G . Then we would see λα λ−1 α −1 = ζ for some ζ ∈ Z (G ), and hence λαn λ−1 = (λα λ−1 )n = ζ n αn = αn . This means that αn is fixed by the action of the Weyl group W (G ) = N T G / T G . Consequently the infinite set {(αn )k | k ∈ Z} could be contained W (G ) W (G ) . This is, however, a finite set, since T G / Z (G ) is an elementary abelian 2-group [7, Remark 1.5]. This in the set T G contradiction completes the proof. 2 ∧ Here we first recall the kernel of a map f : B L → ( B G )∧ p [9,5], and [16], where X p denotes the p-completion of a space X . Let T (or T L ) be a maximal torus of the Lie group L. Suppose N p T denotes the inverse image in the normalizer of a maximal torus T of a p-Sylow subgroup W p of the Weyl group of L. We define a subgroup N p ∞ T of N p T as follows:
T
Z/ p ∞
NpT
Wp
N p∞ T
Wp
Here Z/ p ∞ ⊂ T is the subgroup of elements whose order is a power of p. The kernel of a map f : B L → ( B G )∧p is defined in [9, §1] as follows:
Ker f = {x ∈ N p ∞ T | f | B x 0}. Here x denotes the subgroup of N p ∞ T generated by x. We note that Ker f is a group. Next we recall subgroups related to the center of connected compact Lie groups, [9]. Let Z ( L ) denote the center of L and let W ( L ) denote its Weyl group. If L is a simply-connected simple Lie group other than the exceptional Lie group G 2 , we define
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Z (L, p) = If L = G 2 , then
Z (L) a maximal torus of L
Z (G 2 , p ) =
if | W ( L )| ≡ 0 mod p , if | W ( L )| ≡ 0 mod p .
Z (G 2 ) the center of the subgroup SU (3) of G 2 a maximal torus of G 2
if p = 2, if p = 3, if p 5.
According to [9, Lemma 2], if a map f : B L → ( B G )∧ p is not null homotopic, then Ker f is included in Z ( L , p ). Lemma 1.2. Let K , L and G be connected compact Lie groups. Suppose L is simple. For a map f : B K × B L → B G, if there is a subgroup S 1 of K × L, but S 1 ⊂ K , such that f | B S 1 0, then f | B L 0. Proof. Since S 1 ⊂ K , we can find (x, y ) ∈ S 1 ∩ Ker( f p∧ ) ⊂ K × L where y is an infinite order element in L. Lemma 1.1
/ Z ( L ). Since (x, y ) ∈ Ker( f p∧ ), we see that (x−1 , y −1 ), (x, λ y λ−1 ) ∈ Ker f . says that there is λ ∈ N T L such that λ y λ−1 y −1 ∈ Consequently we obtain the following:
x, λ y λ−1 · x−1 , y −1 = 1, λ y λ−1 y −1 ∈ Ker f p∧ .
Since λ y λ−1 y −1 ∈ / Z ( L ) and L is simple, there is a prime p such that | W ( L )| ≡ 0 mod p, and the composite map B L → B K × B L → B G → ( B G )∧ p is null homotopic. Therefore f | B L 0 by [9]. 2 Let G and G be connected compact Lie groups with maximal tori T and T , respectively. For a map f : B G → B G, we can find a homomorphism (admissible map), φ : T → T by [2, Theorem 1.1] which makes the following diagram commutative:
H ∗ ( B G ; Q)
H ∗ ( B T ; Q)
f∗
( B φ)∗
H ∗ ( B G ; Q)
H ∗ ( B T ; Q)
Here the vertical maps are injective. It is well known that the rational cohomology H ∗ ( B G , Q) is isomorphic to the ring of invariants H ∗ ( B T ; Q) W (G ) . We define the rank of f : B G → B G, denoted by rank( f ), as the rank of the linear map ( B φ)∗ : H 2 ( B T ; Q) → H 2 ( B T ; Q). This is well defined, since the choice of φ is unique up to conjugation. For a finite covering G → G we see H ∗ ( B G ; Q) ∼ = H ∗ ( B G ; Q). Thus, using a universal finite covering, we may assume that the connected compact Lie groups are products of simply-connected simple Lie groups and a torus when we discuss the rank of f . Theorem 1.1. Let K , L and G be connected compact Lie groups. If the following hold:
μ : B K × B L → B G is a map with μ| B K = α and μ| B L = f , then
(1) rank(μ) rank( f ) + rank(α ). (2) If L is semi-simple, then rank(μ) = rank( f ) + rank(α ). Proof. Recall that if n = rank(G ), then H ∗ ( B T G ; Q) is a polynomial ring generated by n variables of degree 2. So H 2 ( B T G ; Q) is regarded as an n-dimensional vector space over Q. For the map μ, we have the following commutative diagram:
H ∗ ( B G ; Q)
H ∗ ( B T G ; Q)
μ∗
( B φ)∗
H ∗ ( B K × B L ; Q)
H ∗ ( B T K × B T L ; Q)
Let k = rank( K ) and = rank( L ). We notice that the admissible map ( B φ)∗ : H 2 ( B T G ; Q) → H 2 ( B T K × B T L ; Q) is presented by a (k + ) × n matrix. This matrix is decomposed as a k × n matrix and an × n matrix obtained from α ∗ and f ∗ respectively. Consequently rank(μ) rank( f ) + rank(α ). It remains to show rank(μ) rank( f ) + rank(α ), if L is semi-simple. The universal covering L of L is a product group of 1-connected simple Lie groups. Since H ∗ ( B L ; Q) ∼ L ; Q), as far as the rank of a map is concerned, we may assume = H ∗ ( B m L = i =1 L i where each L i is simple. We notice that rank( f | BL i ) is equal to either 0 or rank( L i ). We see that rank( f ) =
rank( f | BL i ). Let J = {i | rank( f | BL i ) = 0} and let L = respectively, such that i
rank( B φ| B T 1 ) = rank( T 1 ) = rank(α ),
j∈ J
L j . We can find subtori T 1 and T 2 (= T L ) of T K and T L
rank( B φ| B T 2 ) = rank( T 2 ) = rank( f ).
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If rank(μ) rank( f ) + rank(α ), then there is a subgroup S 1 of T 1 × T 2 , but S 1 ⊂ T 1 , such that B φ| B S 1 0. Lemma 1.2 implies that f | BL j 0 for some L j . Since rank( f | BL j ) = 0, this contradiction completes the proof. 2 Proof of Theorem 1. First we show the part (1). Let K 0 denote the connected component of the compact Lie group K . Suppose μ : B K 0 × B H → B G is the restriction of a pairing B K × B H → B G with μ| B K 0 = α | B K 0 and μ| B H = Bi. Since H is semi-simple with rank( H ) = rank(G ) and rank(μ) rank(G ), we see that Theorem 1.1 implies rank(α | B K 0 ) = 0. Therefore the map α | B K 0 is null homotopic, and α factors through B π0 K . Next we show the part (2). Let α : B π0 K → B G be the induced map. Applying [10, Corollary 1.2] to a restriction of the pairing B π0 K × B T H → B G, we see that the map α factors through BC G ( T H )0 at any prime p. Since G is connected, we see C G ( T H ) = T H by [1, Proposition 4.25]. Let π = π0 K and T = T H . By Proposition 4 of [13] we know that, since T is abelian, any map B π → B T is induced by a homomorphism. Since π is a finite group, we notice [14, §2] that [ B π , B T ] = ∧ ∧ p [ B π p , B T p ]. Therefore α is induced by a homomorphism. 2 Remark 1.1. If the subgroup H of G in Theorem 1 is not semi-simple, the result does not hold. A counterexample is given as follows: Take a maximal torus T G of G so that rank( T G ) = rank(G ). The multiplication map B T G × B T G → B T G composite with Bi : B T G → B G gives a pairing B T G × B T G → B G. We notice that Bi ∈ ( Bi )⊥ ( B T G , B G ). The monomorphism Bi, however, does not factor through the one-point space B π0 T G . Remark 1.2. The p-completion of a connected compact Lie group is a p-compact group, [6]. Suppose f : X → Y is a monomorphism of p-compact groups. If rank( X ) = rank(Y ) and | W ( X )| = | W (Y )|, then f is an isomorphism. Let i : H → G be as in Theorem 1. Furthermore, the orders of their Weyl groups are the same. Then one can show that H is isomorphic to G. We add that, according to [4, Lemma 4.6], the map f is an isomorphism if and only if the mod p cohomological dimensions of X and Y are the same. Theorem 1.2. Let H be a connected subgroup of a connected compact Lie group G. For the inclusion i : H → G, if K is simple and rank(G ) < rank( K ) + rank( H ), then ( Bi )⊥ ( B K , B G ) = 0. Proof. For
α ∈ ( Bi )⊥ ( B K p∧ , B G ∧p ), let μ be the pairing B K p∧ × B H ∧p → B G ∧p , and let φ be its admissible map: μ
B K p∧ × B H ∧ p
( B T K )∧p × ( B T H )∧p
Bφ
BG∧ p
( B T G )∧p
Since rank(G ) < rank( K ) + rank( H ), then rank(φ) < rank( K ) + rank( H ). Notice that φ| T H is one-to-one. Consequently we can find a subgroup S 1 of K × H , but S 1 ⊂ H , such that μ| B S 1 0. Since K is simple, Lemma 1.2 shows μ| B K 0 and therefore ( Bi )⊥ ( B K , B G ) = 0. 2 2. Classical Lie groups and admissible maps In this section we will consider the pairing problem of classical Lie groups. We prove Theorem 2 and the p-completed version of Theorem 3. Proof of Theorem 2. Suppose α ∈ ( Bi )⊥ (BSU (k), BSU (n)). The pairing μ : BSU (k) × BSU (m) → BSU (n) for this map α is also a pairing for Bi. So we see Bi ∈ (α )⊥ (BSU (m), BSU (n)). Hence, for any subgroup Z/ p of SU (m), it follows that Bi | B Z/ p ∈ (α )⊥ ( B Z/ p , BSU (n)∧p ). The map α : BSU(k)∧p → BSU (n)∧p factors through BC SU(n) (Z/ p )∧p up to homotopy [10, Corollary 1.2]. Choose a prime number p which divides m, and let ζ be a primitive p-th root of unity. We define a scalar matrix [ζ ] = diag(ζ, ζ, . . . , ζ ) ∈ SU (m). We will compute the centralizer C SU(n) (Z/ p ):
and
[ζ ] 0 0
I
A C
B D
A · C
B D
=
[ζ ] A [ζ ] B C
D
A [ζ ] B [ζ ] 0 · = . 0 I C [ζ ] D
Consequently B = 0 and C = 0. Thus C SU(n) (Z/ p ) is a subgroup of U (m) × U (n − m). By Theorem 1 of [12], we see that the pairing μ maps BSU (k) trivially into BU (m). Any map from BSU (k) to BU (n − m) factors through BSU (n − m), since BSU (k) is 3-connected. This implies the desired result. An analogous argument is applicable for the case of symplectic groups. 2
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We recall some arguments of admissible maps for the mod p-cohomology and then for the p-adic K-theory. For connected compact Lie groups G and K with maximal tori T G and T K respectively, suppose H ∗ ( B G ; F p ) ∼ = H ∗ ( B T G ; F p ) W (G ) and H ∗ ( B K ; F p ) ∼ = H ∗ ( B T K ; F p )W ( K ) . We note that H ∗ ( B G ; F p ) is isomorphic to H ∗ ( B T G ; F p )W (G ) , for instance, if p does not divide the order of W (G ). For any map f : B G → B K we have the commutative diagram: φf
H ∗(B T K ; Fp )
H ∗(B K ; Fp )
H ∗(B T G ; Fp )
H ∗(B G ; Fp )
f∗
Here φ = φ f is admissible [3] and [2]; namely for any w ∈ W (G ) we can find w ∈ W ( K ) such that w φ = φ w . Recall that H ∗ ( B T n ; F p ) = F p [t 1 , t 2 , . . . , tn ] is a polynomial ring in n variables of degree 2. Hence the admissible map φ over the Steenrod algebra can be regarded as a rank(G ) × rank( K ) matrix. Next consider, [8] and [18], the admissibility for K-theory with coefficients in Z∧ p , the ring of p-adic integers. Recall that
∧ ∧ ∧ W (G ) . Here note K ( B T n ; Z∧ p ) = Z p [[x1 , x2 , . . . , xn ]], the ring of power series, and that K ( B G ; Z p ) is isomorphic to K ( B T G ; Z p ) ∞ that each xi + 1 is the pullback of the Hoph line bundle over C P via the i-th projection map. For any map f : B G → B K we have the commutative diagram:
φ
K ( B T K ; Z∧ p)
K ( B K ; Z∧ p)
K ( B T G ; Z∧ p)
f∗
K ( B G ; Z∧ p)
Again, [8, Lemma 2.1], for any w ∈ W (G ) we can find w ∈ W ( K ) such that w φ = φ w . We will use the following basic results about the maximal tori and the Weyl groups for SO(n) and SU (n) , in order to
cos θ − sin θ prove Theorem 2.1, the p-completed version of Theorem 3. We note first that SO(2) consists of the matrices , sin θ cos θ
and that
T m = SO(2) × · · · × SO(2) ⊂ SO(2m) ⊂ SO(2m + 1) is a maximal torus in both SO(2m) and SO(2m + 1). Let T be the set of diagonal matrices
⎛
⎞
e i θ1
⎜ ⎜ ⎝
e
⎟ ⎟ ⎠
i θ2
..
. e i θn
θ − sin θ ≡ 0 mod 2π . Then T is a maximal torus of SU (n). The 2 × 2 matrix A = cos is diagonalizable. Let sin θ cos θ iθ
1 i 1 e 0 − 1 . This induces a monomorphism j : SO(n) → SU (n) by j ( B ) = Q B Q −1 where P=√ ∈ U (2). Then P A P = 2 1 −i 0 e −i θ B ∈ SO(n) and
where
n
k=1 θk
⎛
⎞
P
⎜
Q =⎜ ⎝
⎟ ⎟. ⎠
P
..
. P
Since Bi B j for the inclusion i : SO(n) → SU (n), we see that, for n = 2m, the admissible map is given by the following m × (2m − 1) matrix:
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
1
−1
⎞ 1 −1
..
⎟ ⎟ ⎟. ⎟ ⎠
.
1
−1 1
m ∧ W (SO(2m)) , where W (SO(2m)) is generated by the permutation representation ∼ We note that K (BSO(2m); Z∧ p ) = K (B T ; Zp )
of Σm and the following m × m matrix
τ:
K. Ishiguro et al. / Topology and its Applications 160 (2013) 264–272
⎛ ⎜ ⎜ τ =⎜ ⎜ ⎝
269
⎞
−1
⎟ ⎟ ⎟. ⎟ ⎠
−1 1
..
. 1
Namely, the Weyl group W (SO(2m)) permutes the xi ’s together with an even number of changes of signs for ∧ K ( B T m ; Z∧ p ) = Z p [[x1 , x2 , . . . , xm ]].
m ∧ W (SO(2m+1)) , where the Weyl group W (SO(2m + 1)) is ∼ If n = 2m + 1, we see that K (BSO(2m + 1); Z∧ p ) = K (B T ; Zp ) generated by the permutation representation of Σm and the following m × m matrix σ :
⎛ ⎜
⎞
−1
⎟ ⎟. ⎠
1
σ =⎜ ⎝
..
. 1
Namely W (SO(2m + 1)) permutes the xi ’s together with an arbitrary number of changes of signs. Finally recall that W (SU (n)) is isomorphic to the symmetric group Σn . The representation as a subgroup of G L (n − 1, Z∧ p)
n−1 ∼ which makes K (BSU (n); Z∧ ; Z∧p )W (SU(n)) is generated by the permutation representation of Σn−1 together with p ) = K (B T the following (n − 1) × (n − 1) matrix:
⎛
⎞ −1 . .. ⎜ .. ⎟ . ⎜ ⎟. ⎝0 ⎠ 1 −1 0 · · · 0 −1 1
0
n
= 0, then the representation of W (SU (n)) can be regarded
0 1as the permutation
1 0 0 repre0 sentation of Σn . For instance, when n = 4, the transposition (1, 2) corresponds to the matrix 1 0 0 , (2, 3) to 0 0 1 , and 001 010
1 0 −1 (3, 4) to 0 1 −1 , respectively. We will use this convention from now on. In other words, if we let
i =1 xi
0 0 −1
To prove Theorem 3, it suffices to prove the following p-completed version, since for a map p-completion α ∧ p is null homotopic, so is the map α .
α : B G → BSU(n), if the
Theorem 2.1. For the inclusion i : SO(n) → SU (n) with n 3, if f = ( Bi )∧ p and G is a connected compact Lie group, then any map in ∧ f ⊥(B G∧ p , BSU (n) p ) is null homotopic for any prime p.
Proof. First we consider the case n = 2m. For commutative diagram:
α ∈ f ⊥ ( B G ∧p , BSU(2m)∧p ) there is a pairing map μ which gives the following
BSO(2m)∧ p f
μ
∧ BG∧ p × BSO(2m) p
BSU (2m)∧ p
α
BG∧ p
∧ ∧ ∼ k ∧ W (G ) . Consider the admissible map φ which gives Assume rank(G ) = k so that K ( B G ∧ p ; Zp ) = K (B G ; Zp ) = K (B T ; Zp ) the commutative diagram:
K ( B T 2m−1 ; Z∧ p)
K (BSU (2m); Z∧ p)
φ
μ∗
m ∧ K ( B T k ; Z∧ p ) ⊗ K (B T ; Zp )
∧ K ( B G ; Z∧ p ) ⊗ K (BSO(2m); Z p )
We recall that φ can be regarded as a (k + m) × (2m − 1) matrix. If φα is a k × (2m − 1) matrix expressing the admissible map which covers α ∗ and φ f is an m × (2m − 1) matrix expressing the admissible map which covers f ∗ , then the (k + m) × (2m − 1) matrix φ is decomposed as follows:
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φ=
φα φf
.
Recall here the following:
⎛
⎞
−1
1
⎜ ⎜ φf = ⎜ ⎜ ⎝
1 −1
..
⎟ ⎟ ⎟. ⎟ ⎠
. −1
1
1 For any w ∈ W (SO(2m))
Ik 0
0 w
φα φf
we can find w
=
φα φf
∈ W (SU (2m)) such that
· w
where I k denotes the k × k identity matrix. In fact, we will see that, for w ∈ W (SO(2m)), the matrix w ∈ W (SU (2m)) is uniquely determined. Notice here that the column vectors of a matrix expressing an element of W (SU (2m)) consists of 2m − 1 vectors out of the set of 2m vectors {e 1 , e 2 , . . . , e 2m−1 , v } where {e 1 , e 2 , . . . , e 2m−1 } is the standard basis and v = t (−1 − 1 · · · − 1). Using this result and our convention, if w = (i , i + 1), one can show w = (2i − 1, 2i + 1)(2i , 2i + 2). And if w = τ , then w = (1, 2)(3, 4). We will show φα = 0. Let φα is expressed as follows:
⎛a
11
a12
φα = ⎝ ... ak1
.. . ak2
· · · a12m−1 ⎞ .. ⎠ . . · · · ak2m−1
Since φα w = φα for any w ∈ W (SU (2m)), we see that ai j = ai for all 1 j , 2m − 1, and ai1 = −(2m − 1)ai1 so that 2mai1 = 0. Since the p-adic integers Z∧ p is torsion-free, each a i1 must be zero, and hence φα = 0. Consequently, K (μ) = 0. A result of Zabrodsky, [19, Theorem 1], shows that ( Bi )⊥ ( B G , BSU (2m)) = 0. It remains to consider the case n = 2m + 1.
BSO(2m + 1)∧ p f
μ
∧ BG∧ p × BSO(2m + 1) p
BSU (2m + 1)∧ p
α
BG∧ p
Again, for w ∈ W (SO(2m + 1)), one can show that the matrix w ∈ W (SU (2m + 1)) is uniquely determined. If w = (i , i + 1), then w = (2i − 1, 2i + 1)(2i , 2i + 2). And if w = σ , then w = (1, 2). Similarly, we can show φα = 0, and hence ( Bi )⊥ ( B G , BSU (2m + 1)) = 0. 2 Remark 2.1. The Lie group SU (n) is a subgroup of the symplectic group Sp(n). Here we consider what if SU (n) is replaced by Sp(n) in Theorem 3. It seems that the things are different. First, in the proof of the theorem, a property of admissible maps has been important. Namely, for any w ∈ W (SO(n)), we see that w ∈ W (SU (n)) is uniquely determined. In the case of the inclusion j : SO(n) → Sp(n), this property no longer holds. For instance, if f = ( B j )∧ p , then φ f = (1, −1, 0). We recall
that W (SO(3)) = {±1} and W (Sp(3)) = Σ3 ,
−1 0 0 0 10 0 01
. Take w = −1. Then w can be either − I 3 or
0 1 0 100 001
. Moreover, for
− I 2 , four matrices can be w . Apparently our proof for SU(n) is not applicable for Sp(n). In fact we will see
n = 4, if w = ( B j )⊥ ( B S 1 , BSp(n)) = 0. To see this, consider the homomorphisms of Lie groups:
SO(n) × S 1
U (n) × Z (U (n))
U (n)
Sp(n)
K. Ishiguro et al. / Topology and its Applications 160 (2013) 264–272
271
where the map SO(n) → U (n) is the canonical inclusion and S 1 → Z (U (n)) is the identity map. We can see that the axis B S 1 → BSp(n) of the induced map BSO(n) × B S 1 → BSp(n) is an essential map. Finally we consider the mod p-cohomology version. The result is slightly different from the K-theory case. Theorem 2.2. Let i : SO(n) → SU (n) be the inclusion for n 3. Suppose that a space X is a connected p-compact group with maximal torus T X and Weyl group W ( X ) such that the mod p cohomology H ∗ ( B X ; F p ) is isomorphic to the ring of invariants H ∗ ( B T X ; F p ) W ( X ) . Suppose the following diagram is commutative over the Steenrod algebra A p :
H ∗ (BSO(n); F p ) ( Bi )∗ ψ
H ∗ (BSU (n); F p )
H ∗ ( B X ; F p ) ⊗ H ∗ (BSO(n); F p ) β
H ∗(B X ; Fp ) Then rank(φβ ) 1. Moreover, β = 0 if p does not divide n. Proof. The argument here is similar to the one used in the proof of Theorem 3. First we treat the case that a prime p is odd and n = 2m. Assume rank( X ) = k so that H ∗ ( B X ; F p ) ∼ = H ∗ ( B T k ; F p )W ( X ) . We notice that H ∗ (BSO(2m); F p ) ∼ = H ∗ ( B T m ; F p )W (SO(2m)) and H ∗ (BSU (2m); F p ) ∼ = H ∗ ( B T 2m−1 ; F p )W (SU(2m)) , since p is odd. Consider the admissible map φ which gives the following commutative diagram:
H ∗ ( B T 2m−1 ; F p )
φ
H ∗ (BSU (2m); F p )
ψ
H ∗(B T k ; Fp ) ⊗ H ∗(B T m; Fp )
H ∗ ( B X ; F p ) ⊗ H ∗ (BSO(2m); F p )
We note here that the (k + m) × (2m − 1) matrix φ is decomposed as follows:
φ=
φβ
.
φ( Bi )∗
As a subgroup of G L (2m, F p ), the Weyl group W (SO(2m)) is generated by the permutation representation of Σm and the following m × m matrix:
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
−1
⎞
⎟ ⎟ ⎟. ⎟ ⎠
−1 1
..
. 1
For φβ = (ai j ), as seen in the proof of Theorem 3, it follows that ai j = ai for all 1 j , 2m − 1, and ai1 = −(2m − 1)ai1 . Thus we see that rank(φβ ) 1. Notice that 2mai1 = nai1 = 0. If p does not divide n, then each ai1 is zero. Therefore φβ = 0, and hence β = 0. The case of n = 2m + 1 is similarly proved. Suppose next p = 2. Doubling the degree, we see that the A2 -algebra structure of H ∗ (BSO(n); F2 ) can be same as that of H ∗ (BSU (n); F2 ). Namely, if we express H ∗ (BSO(n); F2 ) = F2 [x2 , x3 , . . . , xn ], then H ∗ (BSU (n); F2 ) is isomorphic to F2 [x22 , x23 , . . . , xn2 ]. For n 3, notice that H ∗ (BSU (n); F2 ) ∼ = H ∗ ( B T n−1 ; F2 )W (SU(n)) . Consider the admissible map φ which gives the commutative diagram:
H ∗ ( B T n −1 ; F p )
H ∗ (BSU (n); F p )
φ
H ∗ ( B T k ; F p ) ⊗ H ∗ ( B T n −1 ; F p )
ψ
Im ψ
The (k + m) × (n − 1) matrix φ is decomposed as follows:
φ=
φβ I n −1
.
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K. Ishiguro et al. / Topology and its Applications 160 (2013) 264–272
Since φα w = φα for any w ∈ W (SU (n)), we see that rank(φβ ) 1. We see also that if n is odd, then β = 0. This completes the proof. 2 Remark 2.2. In the case n ≡ 0 mod p, there is an A p -map
H ∗ BSU (n); F p → H ∗ ( B X ; F p ) ⊗ H ∗ BSO(n); F p
∗ ∗ for B X = ( B S 1 )∧ p such that the restriction H (BSU (n); F p ) → H ( B X ; F p ) is non-zero. Let ζ ∈ C be a primitive p-th root of unity so that ζ n = 1. A homomorphism ϕ : SO(n) × Z/ p → SU (n) is induced by ϕ ( A , ζ ) = A · (ζ I n ), and is delooped to the following map:
B ϕ : BSO(n) × B Z/ p → BSU (n). This gives an A p -map
H ∗ BSU (n); F p → H ∗ ( B Z/ p ; F p ) ⊗ H ∗ BSO(n); F p
which factors through H ∗ ( B S 1 ; F p ) ⊗ H ∗ (BSO(n); F p ) for dimensional reason. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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