Pairing problem of classifying spaces and admissible maps
Kenshi Ishiguro and Fumihisa Yayama
1ʢ Received November 30, 2013 ʣ
Abstract This is a survey about the pairing problem of classifying spaces and some properties of admissible maps, based mainly on [12], [13] and [16].
We consider the maps between classifying spaces of compact Lie groups of the form BK × BL −→ BG . In § 1, the restriction map BL −→ BG is a weak epimorphism. And, in § 2, the map is replaced by a monomorphism. The proofs make use of admissible maps, which can be discussed in § 3. We will overview some historical background as well as recent developments of our subjects.
AMS Classification 55R37; 55R35, 55P60
Keywords pairing, classifying space, admissible map, Lie group, p –compact group, invariant ring
We consider the pairing problem of classifying spaces of compact Lie groups as well as p –compact groups, [12], [13], [22] and [16]. A p –compact group is a p –local generalization of a compact Lie group G. Its p–completion G
∧pis a p –compact group if π
0(G) is a p –group, [5] and [20].
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 , [24]. We warn that, given α and f , a pairing µ need not be unique. For instance, if X = Y = S
1and Z = K( Z , 2), there is a pairing µ ̸ = 0 such that µ |
X≃ 0 ≃ µ |
Y. Here we give a necessary and sufficient condition that a map α : X −→ Z be contained in f
⊥(X, Z) in terms of mapping spaces. Then a special case of classifying space version is discussed.
Theorem 1 ([12]) Suppose X, Y and Z are pointed connected spaces. For a map f : Y −→ Z , a map α : X −→ Z is contained in f
⊥(X, Z ) if and only if the map f factors through map(X, Z )
α, the connected component of the mapping space containing α , under the evaluation map map(X, Z )
α−→ Z .
1
Department of Applied Mathematics, Fukuoka University, Fukuoka, 814-0180,
Japan
Y
map(X, Z )
α� Z µ
f
�� �� ��� � ev
�
It is easy to see that α ∈ f
⊥(X, Z ) if and only if f ∈ α
⊥(Y, Z ). Consequently α must factor through map(Y, Z )
f. If Z is an H–space, then f
⊥(X, Z ) = [X, Z ] : for the H –multiplication m : Z × Z −→ Z , a pairing of f and α is given by the composite map m ◦ (α × f ). It follows that, for example, if G, K , and L are compact Lie groups, we see f
⊥(K, G) = [K, G] with f : L −→ G . In this paper we will study the classifying space version, namely f
⊥(BK, BG) with f : BL −→ BG. As group theoretical analog indicates, our results will show that few maps in [BK, BG] belong to f
⊥(BK, BG) , in general. Other forms of axial maps (pairings) are studied, for example, in [9], [10] and [26].
In the case of classifying spaces [12], we have the following commutative dia- gram:
BK × BL
BK BL
� BG f
µ
�� α �� � �
�� ��� �
�
�
It is worth to recall some properties of homomorphisms. 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 ρ(L) must be
contained in the centralizer of α in G, denoted by C
G(α). The following
is a “BG”–analog at a prime p . If a map α : BK −→ BG is induced by a
homomorphism, let C
G(α) denote the centralizer of the homomorphism. For
a p –toral group K (a group extension of a torus by a finite p –group), it is
known, [7] and [22], that any map α : BK −→ BG (at p ) has the form α = Bη
(α = (Bη)
∧p) for some homomorphism η . Let BG
∧pdenote the p –completion
of BG. Since map(BK, BG)
αis p –equivalent to BC
G(α), the following is
immediate from Theorem 1.
Corollary 1 ([12]) Suppose K is p –toral. Then α ∈ f
⊥(BK
p∧, BG
∧p) if and only if the map f factors through BC
G(α)
∧pup to homotopy, under the map induced by the inclusion BC
G(α)
∧p−→ BG
∧p.
BL
∧pBC
G(α)
∧pBG
∧p� µ
f
�� �� �� � ev
�
As the above indicates, if the mapping space is computable, then the set of the homotopy classes of axes f
⊥(X, Z ) would be determined. It is, however, hard to compute map(X, Z )
αor map(Y, Z )
fin general. Thus our work is to consider the problem, sometimes without calculation of mapping spaces.
1 Pairing problem for weak epimorphisms
First we consider f
⊥(BK, BG) when f : BL −→ BG is a weak epimorphism, defined as follows. Suppose L and G are connected. A map BL −→ BG or BL
∧p−→ BG
∧pis called a weak epimorphism, [14], if there exists a fibration Z −→ BL −→ BG or Z −→ BL
∧p−→ BG
∧psuch that H
∗(ΩZ ; Q ) is a finite dimen- sional Q –module or that H
∗(ΩZ ; Z
∧p) ⊗ Q is a finite dimensional Q
∧p–module, respectively. An obvious example of a weak epimorphism is given by a map f = Bρ induced by a group epimorphism ρ. The unstable Adams operations { ψ
k} are also weak epimorphisms. For any ψ
kon BU (n), one can show that (ψ
k)
⊥(BS
1, BU (n)) = Z using the following result.
Theorem 2 ([12]) Let L and G be connected compact Lie groups and let K be a compact (not necessarily connected) Lie group. If f : BL −→ BG is a weak epimorphism, the following hold:
(1) If α ∈ f
⊥(BK, BG), then the map α factors through BZ (G) up to homo- topy, where Z(G) denotes the center of G.
(2) Moreover, we have f
⊥(BK, BG) = Hom(K, Z(G)) .
Furthermore, if we take K = L = G and f = α = id, the problem now asks whether BG is an H–space. Theorem 2 implies that BG is an H–space if and Y
map(X, Z )
α� Z µ
f
�� �� ��� � ev
�
It is easy to see that α ∈ f
⊥(X, Z ) if and only if f ∈ α
⊥(Y, Z). Consequently α must factor through map(Y, Z )
f. If Z is an H–space, then f
⊥(X, Z ) = [X, Z ] : for the H –multiplication m : Z × Z −→ Z , a pairing of f and α is given by the composite map m ◦ (α × f ). It follows that, for example, if G, K , and L are compact Lie groups, we see f
⊥(K, G) = [K, G] with f : L −→ G . In this paper we will study the classifying space version, namely f
⊥(BK, BG) with f : BL −→ BG. As group theoretical analog indicates, our results will show that few maps in [BK, BG] belong to f
⊥(BK, BG) , in general. Other forms of axial maps (pairings) are studied, for example, in [9], [10] and [26].
In the case of classifying spaces [12], we have the following commutative dia- gram:
BK × BL
BK BL
� BG f
µ
�� α �� � �
�� ��� �
�
�
It is worth to recall some properties of homomorphisms. 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 ρ(L) must be
contained in the centralizer of α in G, denoted by C
G(α). The following
is a “BG”–analog at a prime p . If a map α : BK −→ BG is induced by a
homomorphism, let C
G(α) denote the centralizer of the homomorphism. For
a p–toral group K (a group extension of a torus by a finite p –group), it is
known, [7] and [22], that any map α : BK −→ BG (at p ) has the form α = Bη
(α = (Bη)
∧p) for some homomorphism η . Let BG
∧pdenote the p –completion
of BG. Since map(BK, BG)
αis p –equivalent to BC
G(α), the following is
immediate from Theorem 1.
only if G is abelian. ( G need not be connected.) So G is a product group of a torus and a finite abelian group. A p –local version is also available. Namely, if BG
∧pis an H–space, then G must be a product group of a torus and a finite p – nilpotent group with an abelian p –Sylow group. Related results were obtained in [3] and [19].
Here consider f
⊥(BX, BZ ) for a map f : BY −→ BZ of p–compact groups. We first recall some basic things about the p –compact groups and pairing problems, and then state our results. A p –compact group, [5], is a loop space X such that X is F
p–finite and that its classifying space BX is F
p–complete. As mentioned before, the p –completion of a compact Lie group G is a p –compact group if π
0(G) is a p –group. For an odd dimensional sphere S
2n−1, it is known that its p –completion has a loop structure if n divides p − 1. This is an example of p –compact groups other than compact Lie groups. More examples are known as Clark–Ewing p –compact groups, [4].
For p –compact groups X and Y , a pointed map f : BX −→ BY is called a homomorphism. Let Y /X denote the homotopy fibre of f . The homomorphism f is called a monomorphism if Y /X is F
p–finite, and an epimorphism if the loop space Ω(Y /X) is a p –compact group.
The centralizer of f is the loop space of the component containing f of the mapping space of unpointed maps, denoted by Ωmap(BX, BY )
f. A homomor- phism is called central if the evaluation map, ev : map(BX, BY )
f−→ BY , is a homotopy equivalence According to [6], any p –compact group X has a unique maximal central subgroup that is called the center of X and denoted by C(X ).
It is also shown in [6] that BC (X ) ≃ map(BX, BX )
idwhere id : BX −→ BX is the identity homomorphism.
Suppose that X , Y and Z are p–compact groups, and that α : BX −→ BZ and f : BY −→ BZ are homomorphisms. The homotopy class of α is said to be contained in the set of the homotopy classes of axes f
⊥(BX, BZ) if there is a map (called a pairing) µ : BX × BY −→ BZ with restrictions (axes) µ |
BX≃ α and µ |
BY≃ f . Of course, if α ∈ f
⊥(BX, BZ), we have the following homotopy commutative diagram:
BX × BY
BX BY
� BZ f µ
�� α �� � �
�� ��� �
�
�
For a weak epimorphism f of the classifying spaces of connected compact Lie groups, the set of the homotopy classes of axes has been determined. We obtain analogous results for p –compact groups.
Theorem 3 ([13]) Suppose X is a p–compact group. If either (i) f : BY −→ BZ is an epimorphism of p –compact groups, or
(ii) f : BY −→ BZ is a homomorphism of connected p –compact groups such that H
∗(Ω(Z/Y ); Z
∧p) ⊗ Q is a finite dimensional Q
∧p–vector space
then the following hold:
(a) If α ∈ f
⊥(BX, BZ ) , then the map α factors through the classifying space of the center of Z , denoted by C(Z) , up to homotopy.
(b) Moreover, we have f
⊥(BX, BZ ) = [BX, BC(Z )] .
There is a strong relationship between pairing problems and mapping spaces.
The following result shows that, for the homomorphism f : BY −→ BZ in Theo- rem 3, no p –compact groups find a difference between BC (Z) and map(BY, BZ )
f.
Corollary 2 ([13]) Let f : BY −→ BZ be as in Theorem 3. For any p – compact group X , the map of homotopy sets
[BX, BC(Z )] −→ [BX, map(BY, BZ )
f]
is bijective, where the above map is induced by the canonical map BC (Z) = map(BZ, BZ)
id−→ map(BY, BZ)
f.
Again, if we take X = Y = Z and f = α = id, the problem asks whether BX is an H–space. Theorem 3 implies that BX is an H–space if and only if X is mod p equivalent to a product space of a torus and a finite abelian p –group.
2 Pairing problem for monomorphisms
Next we will consider the problem for a map which is not a weak epimorphism.
As a test map, we take the map BSU (n) −→ BU (n) induced from the inclusion i : SU (n) −→ U (n).
only if G is abelian. (G need not be connected.) So G is a product group of a torus and a finite abelian group. A p–local version is also available. Namely, if BG
∧pis an H–space, then G must be a product group of a torus and a finite p – nilpotent group with an abelian p –Sylow group. Related results were obtained in [3] and [19].
Here consider f
⊥(BX, BZ ) for a map f : BY −→ BZ of p–compact groups. We first recall some basic things about the p –compact groups and pairing problems, and then state our results. A p –compact group, [5], is a loop space X such that X is F
p–finite and that its classifying space BX is F
p–complete. As mentioned before, the p –completion of a compact Lie group G is a p –compact group if π
0(G) is a p –group. For an odd dimensional sphere S
2n−1, it is known that its p–completion has a loop structure if n divides p − 1. This is an example of p –compact groups other than compact Lie groups. More examples are known as Clark–Ewing p –compact groups, [4].
For p –compact groups X and Y , a pointed map f : BX −→ BY is called a homomorphism. Let Y /X denote the homotopy fibre of f . The homomorphism f is called a monomorphism if Y /X is F
p–finite, and an epimorphism if the loop space Ω(Y /X ) is a p –compact group.
The centralizer of f is the loop space of the component containing f of the mapping space of unpointed maps, denoted by Ωmap(BX, BY )
f. A homomor- phism is called central if the evaluation map, ev : map(BX, BY )
f−→ BY , is a homotopy equivalence According to [6], any p –compact group X has a unique maximal central subgroup that is called the center of X and denoted by C(X ).
It is also shown in [6] that BC (X) ≃ map(BX, BX )
idwhere id : BX −→ BX is the identity homomorphism.
Suppose that X , Y and Z are p–compact groups, and that α : BX −→ BZ and f : BY −→ BZ are homomorphisms. The homotopy class of α is said to be contained in the set of the homotopy classes of axes f
⊥(BX, BZ) if there is a map (called a pairing) µ : BX × BY −→ BZ with restrictions (axes) µ |
BX≃ α and µ |
BY≃ f . Of course, if α ∈ f
⊥(BX, BZ), we have the following homotopy commutative diagram:
BX × BY
BX BY
� BZ f µ
�� α �� � �
�� ��� �
�
�
BK × BSU (n)
BK BSU (n)
BU (n)
� Bi µ
�� �� α � �
�� �� � �
�
�
The following will indicate that the group theoretical analog also holds for some maps other than weak epimorphisms.
Theorem 4 ([15]) For the inclusion i : SU (n) −→ U (n), if a connected com- pact Lie group K is semi–simple, then any map in (Bi)
⊥(BK, BU (n)) is null homotopic:
(Bi)
⊥(BK, BU (n)) = 0
Corollary 3 ([15]) Let Z (U (n)) denote the center of U (n) . Then the fol- lowing hold:
(1) If α ∈ (Bi)
⊥(BU (k), BU (n)), the map α factors through BZ (U (n)) up to homotopy.
(2) Moreover, we have (Bi)
⊥(BU (k), BU (n)) = Hom(U (k), Z(U (n))) .
Theorem 4 is a consequence of the following proposition, which turns out to be the mod p–cohomology version.
Proposition 1 ([15]) Let i : SU (n) −→ U (n) be the natural inclusion. Sup- pose that for an odd prime p a space X is a connected p–compact group with maximal torus T
Xand Weyl group W (X) such that the mod p cohomology H
∗(BX ; F
p) is isomorphic to the ring of invariants H
∗(BT
X; F
p)
W(X). If f = (Bi)
∧pand α ∈ f
⊥(BX, BU (n)
∧p) , then α
∗: H
∗(BU (n); F
p) −→ H
∗(BX ; F
p) factors through H
∗(BZ (U (n)); F
p) over the Steenrod algebra.
H
∗(BZ (U (n)); F
p)
H
∗(BU (n); F
p) � H
∗(BX ; F
p)
�
α
∗� �
�
�
Replacing the inclusion SU (n) −→ U (n) by SO(n) −→ SU (n), we obtain an ana-
logus result. We note, however, that the group K need not be semi–simple.
Theorem 5 ([16]) For the inclusion i : SO(n) −→ SU (n) with n ≥ 3 , if K is a connected compact Lie group, then any map in (Bi)
⊥(BK, BSU(n)) is null homotopic:
(Bi)
⊥(BK, BSU (n)) = 0
In fact, the p –completed version of this result has been shown in [16]. So the above theorem is its easy consequence. The Lie group SU (n) is a sub- group of the symplectic group Sp(n). According to [16, Remark 2.1], if SU (n) is replaced by Sp(n), the corresponding result does not hold. A counter- example is given by the fact that (Bj)
⊥(BS
1, BSp(n)) ̸ = 0 for the inclusion j : SO(n) −→ Sp(n).
The following results are some applications of admissible maps which will be discussed in the next section.
Theorem 6 ([16]) Suppose that K is a compact Lie group, and that a con- nected 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)
⊥(BK, BG), then the following hold:
(1) The map α : BK −→ BG factors through Bπ
0K up to homotopy under the map induced by the projection q : K −→ π
0K . In particular, if K is connected, the map α is null homotopic.
(2) There is a homomorphism ρ : π
0K −→ G such that α ≃ Bρ ◦ Bq , and the image of the homomorphism ρ(π
0K ) is contained in the centralizer C
G(H) .
BK × BH
BK BH
Bπ
0K
� BG
� Bi µ
α Bq
�� �� � �
�� �� � �
�
� �
The condition on H being semi–simple is necessary. Some counter-examples can be found in [16, Remark 1.1] and [27].
Theorem 7 ([16]) 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) (Bj)
⊥(BSp(k), BSp(n)) = [BSp(k), BSp(n − m)] . BK × BSU (n)
BK BSU (n)
BU (n)
� Bi µ
�� �� α � �
�� �� � �
�
�
The following will indicate that the group theoretical analog also holds for some maps other than weak epimorphisms.
Theorem 4 ([15]) For the inclusion i : SU (n) −→ U (n), if a connected com- pact Lie group K is semi–simple, then any map in (Bi)
⊥(BK, BU (n)) is null homotopic:
(Bi)
⊥(BK, BU (n)) = 0
Corollary 3 ([15]) Let Z(U (n)) denote the center of U (n) . Then the fol- lowing hold:
(1) If α ∈ (Bi)
⊥(BU (k), BU (n)), the map α factors through BZ (U (n)) up to homotopy.
(2) Moreover, we have (Bi)
⊥(BU (k), BU (n)) = Hom(U (k), Z (U (n))) .
Theorem 4 is a consequence of the following proposition, which turns out to be the mod p–cohomology version.
Proposition 1 ([15]) Let i : SU (n) −→ U (n) be the natural inclusion. Sup- pose that for an odd prime p a space X is a connected p–compact group with maximal torus T
Xand Weyl group W (X ) such that the mod p cohomology H
∗(BX ; F
p) is isomorphic to the ring of invariants H
∗(BT
X; F
p)
W(X). If f = (Bi)
∧pand α ∈ f
⊥(BX, BU (n)
∧p) , then α
∗: H
∗(BU (n); F
p) −→ H
∗(BX ; F
p) factors through H
∗(BZ (U (n)); F
p) over the Steenrod algebra.
H
∗(BZ (U (n)); F
p)
H
∗(BU (n); F
p) � H
∗(BX ; F
p)
�
α
∗� �
�
�
Replacing the inclusion SU (n) −→ U (n) by SO(n) −→ SU (n), we obtain an ana-
logus result. We note, however, that the group K need not be semi–simple.
3 Admissible maps and invariant rings
We recall some arguments of admissible maps for the rational cohomology, [1]. It is well–known that, for a connected compact Lie group G, the ra- tional cohomology H
∗(BG, Q ) is isomorphic to the ring of invariants under the action of the Weyl group W (G). Consequently, for connected compact Lie groups G and K with maximal tori T
Gand T
Krespectively, we see H
∗(BG; Q ) ∼ = H
∗(BT
G; Q )
W(G)and H
∗(BK; Q ) ∼ = H
∗(BT
K; Q )
W(K). For any map f : BG −→ BK we have the commutative diagram:
H
∗(BT
K; Q ) −−−−→
ϕfH
∗(BT
G; Q )
�
�
H
∗(BK; Q ) −−−−→
f∗
H
∗(BG; Q )
Here ϕ = ϕ
fis admissible ; namely for any w ∈ W (G) we can find w
′∈ W (K ) such that wϕ = ϕw
′. We warn, however, that w
′need not be uniquely determined unless the admissible map is regular.
Theorem 8 ([1], Theorem 2.21) There exists a homomorphism ρ : W (G) −→ W (K ) such that wϕ = ϕρ(w).
Recall that H
∗(BT
n; Q ) = Q [t
1, t
2, · · · , t
n] is a polynomial ring in n variables of degree 2. Hence the admissible map ϕ can be regarded as a rank(G) × rank(K ) matrix, since the ring homomorphism is determined by a linear map on the vector space H
2(BT
K; Q ). Notice, [2], that the rational cohomology can be replaced by the mod p –cohomology when p is large. We note that H
∗(BG; F
p) is isomorphic to H
∗(BT
G; F
p)
W(G), for instance, if p does not divide the order of W (G). As mentioned in § 2, the proof of Theorem 5 uses the fact that, for any w ∈ W (SO(n)), we can find w
′∈ W (SU (n)) uniquely. The admissible map for BSO(2m + 1) −→ BSU (2m + 1) is given by the following m × 2m matrix:
1 − 1
1 − 1 . ..
1 − 1
1 − 1
If SU (n) is replaced by the symplectic group Sp(n), the property of uniqueness no longer holds, [16, Remark 2.1].
The admissible self–maps for H
∗(BU (n); F
p) ∼ = H
∗(BT
n; F
p)
Σnare as follows:
ϕ =
(
a1 a2 ··· an· · ·
· · ·
· · ·
a1 a2 ··· an
) or
(
a b ··· bb a··· b
······
b b ··· a
)
The admissible map for ϕ
kfor the unstable Adams operations { ψ
k} on BU (n) is the following scalar matrix:
ϕ
k=
k 0
. ..
0 k
It is known that { ψ
k} on BG, in general, exsits if and only if (k, | W (G) | ) = 1, [8], [25] and [11].
There are other admissible maps that are not realizable as maps between clas- sifying spaces. The Weyl group W (G
2) of the exceptional Lie group G
2is the dihedral group of order 12 presented as D
12=< r, s | r
6= s
2= 1, srs = r
5>.
The matrix (integral) representation can be taken as follows:
r =
( 1 − 1
1 0
)
and s =
( 1 − 1 0 − 1
) .
Proposition 2 The map ϕ =
( 2 − 1 1 − 2
)
is admissible, and it is not realizable as a map on BG
2.
Proof The map ϕ is admissible, since ϕrϕ
−1= r
−1and ϕsϕ
−1= sr
−1. The admissible map is not realizable, since ϕ
2=
( 3 0 0 3
)
= ψ
3and the order of W (G
2) is divisible by 3.
The Weyl group W (F
4) of the exceptional Lie group F
4is generated by the following four reflections:
w
1=
1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1
, w
2=
1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
3 Admissible maps and invariant rings
We recall some arguments of admissible maps for the rational cohomology, [1]. It is well–known that, for a connected compact Lie group G, the ra- tional cohomology H
∗(BG, Q ) is isomorphic to the ring of invariants under the action of the Weyl group W (G). Consequently, for connected compact Lie groups G and K with maximal tori T
Gand T
Krespectively, we see H
∗(BG; Q ) ∼ = H
∗(BT
G; Q )
W(G)and H
∗(BK; Q ) ∼ = H
∗(BT
K; Q )
W(K). For any map f : BG −→ BK we have the commutative diagram:
H
∗(BT
K; Q ) −−−−→
ϕfH
∗(BT
G; Q )
�
�
H
∗(BK; Q ) −−−−→
f∗
