Nilpotent $ n $-tuples in $ SU (2) $

NILPOTENT n-TUPLES IN SU(2)

arXiv:1611.05937v2 [math.AT] 1 Mar 2017

Omar Antol´ın Camarena and Bernardo Villarreal∗

Abstract: Let Fn /Γqn denote the free q-nilpotent group on n-generators. We describe the connected components of the spaces of homomorphisms Hom(Fn /Γqn , SU (2)) arising from non-commuting q-nilpotent n-tuples. We prove this by showing that all these n-tuples are conjugated to n-tuples consisting of elements in the the generalized quaternion groups Q2q ⊂ SU (2), of order 2q . Using this result, we exhibit the homotopy type of ΣHom(Fn /Γqn , SU (2)) and a homotopy description of the classifying spaces B(q, SU (2)) of transitionally q-nilpotent principal SU (2)-bundles. The above computations are also done for SO(3) and U (2). We also include cohomology calculations for the spaces B(r, Q2q ) for low values of r.

Contents 1 2 3 4 5 6 7 A

Introduction Non-abelian nilpotent subgroups of SU (2) Consequences for SO(3) and U (2) 2-nilpotent tuples in U (m) Stable Homotopy type Cohomology of B(r, Q2q ) for r = 2 and 3 Homotopy type of B(q, SU (2)) Computing annihilators in Singular

1

1 3 8 10 11 15 18 19

Introduction

Spaces of homomorphisms have been of interest for a long time now. One of the basic problems is to compute the number of connected components. Given a finitely generated group Γ on k-generators and a topological group G, the set of homomorphisms Hom(Γ, G) is in one to one correspondence with the subset of the product Gk consisting of k-tuples satisfying the relations that hold among the generators; we topologize Hom(Γ, G) as a subspace of Gk , the resulting space is independent of the chosen generators. In this paper we mainly focus on the spaces of homomorphisms of the finitely generated free q-nilpotent groups Fn /Γqn on the Lie group SU(2). Fn stands for the free group on n-generators and Γqn := Γq (Fn ), the q-th stage of the descending central series of Fn . As noted in [9] the case q = 3 is the same as computing the space of almost commuting tuples of SU(2) which were completely described ∗

Supported by a CONACyT fellowship.

1

O. Antol´ın Camarena and B. Villarreal

2

in [2]. We generalize this for q-nilpotent tuples with q ≥ 4. We prove that all non-abelian nilpotent subgroups of SU(2) are conjugated to the quaternion group Q8 or to one of the generalized quaternions Q2q of order 2q . Using this we prove the following. Theorem 1.1. Let q ≥ 3 and n ≥ 2. Then Hom(Fn /Γqn , SU(2)) ∼ = Hom(Zn , SU(2)) ⊔ where C(n, q) =

G

P U(2)

C(n,q)

2n−2 (2n − 1)(2n−1 − 1) + (2n − 1)(2(q−3)(n−1) − 1)22n−3 . 3

In section 3, we study the consequences of Theorem 1.1 for the q-nilpotent tuples of SO(3) and U(2). The double covering map π : SU(2) → SO(3) induces a surjective map Hom(Fn /Γqn , SU(2)) → Hom(Fn /Γnq−1 , SO(3)), which we can use to compute the connected components of Hom(Fn /Γqn , SO(3)). The case q = 2 was originally proved in [8]. Corollary 1.2. Let q ≥ 3, n ≥ 2 and Hom(Zn , SO(3))1 be the connected component that contains the identity n-tuple (I, . . . , I). Then G G S 3 /Q8 ⊔ S 3 /C4 , Hom(Fn /Γqn , SO(3)) ∼ = Hom(Zn , SO(3))1 ⊔ M (n,2)

n

M (n,q)

n−1

where M(n, 2) = (2 −1)(23 −1) , M(n, q) = (2n − 1)(2(q−3)(n−1) − 1)2n−2 and C4 is a cyclic group of order 4. Both C4 and Q8 act by conjugation on S 3 . The group homomorphism S 1 ×SU(2) → U(2) given by (λ, X) 7→ λX induces a surjective map (S 1 )n × Hom(Fn /Γqn , SU(2)) → Hom(Fn /Γqn , U(2))

and modding out the left hand side by the natural diagonal action of (Z/2)n we get a homeomorphism. Corollary 1.3. Let q ≥ 3 and n ≥ 2, then G G (S 1 )n ×(Z/2) P U(2), (S 1 )n ×(Z/2)2 P U(2)) ⊔ Hom(Fn /Γqn , U(2)) ∼ = Hom(Zn , U(2)) ⊔ f(n,3) M

f(n, q) = M(n, q − 1) is as in the previous Corollary. where M

f(n,q) M

Section 4 contains a description of the space of representations Rep(Fn /Γ3 , U(m)) (the orbit space associated to the conjugation action of U(m) over Hom(Fn /Γ3n , U(m))) that expresses it as a union of almost commuting tuples of block matrix subgroups of U(m) that can be identified with the direct product U(m1 ) × · · · × U(mk ) where m1 + · · · + mk = m.

Nilpotent n-tuples in SU (2)

3

In section 5, using the homotopy stable decomposition obtained in [9] for the spaces Hom(Fn /Γqn , G) (for G a real algebraic linear group), we compute the stable homotopy type of the spaces Hom(Fn /Γqn , SU(2)) and Hom(Fn /Γqn , SO(3)) after one suspension. The spaces Hom(Fn /Γqn , G) assemble into a simplicial space and we denote its geometric realization as B(q, G). These spaces were originally introduced in [1]. Using our work in section 1, and [1, Theorem 4.6] applied to Q2q we get that q−r 2q−2 2 r q B(r, Q2 ) ≃ B ∗Z/2r−1 Q2 ∗Z/2 ∗Z/2 Z/4

where ∗A denotes the amalgamated product along A. The above formula allowed us to do some computations in cohomology for q = 2. In this case, the space B(2, G) is denoted by Bcom G.

Proposition 1.4. H ∗ (Bcom Q8 ; F2 ) ∼ = F2 [y1 , y2 , y3, z]/(yi yj , y12 + y22 + y32 , i 6= j) where yi has degree 1 and z degree 2. Let q ≥ 4. Then the F2 -cohomology ring of Bcom Q2q is given by H ∗ (Bcom Q2q ; F2 ) ∼ = F2 [x1 , x2 , y1 , . . . , y2q−2 , z]/(x21 , xk yi , yi yj , i 6= j, x2 +

q−2 2 X

yi2),

i=1

where x1 , yj have degree 1 and x2 , z have degree 2. Increasing the degree of nilpotency makes cohomology calculations considerably harder as we show it for the space B(3, Q16 ) (we verified our calculations using Singular). In the last section, we describe the homotopy type of the classifying spaces B(q, SU(2)) as the homotopy pushouts P U(2) ×Nq B(q − 1, Q2q ) /

B(q − 1, SU(2))

P U(2) ×Nq BQ2q

/

B(q, SU(2)),

where Nq is the normalizer of the dihedral group D2q−1 in P U(2). We also show two interesting principal BZ/2-bundles, namely B(q, SU(2)) → B(q − 1, SO(3)) and BS 1 × B(q, SU(2)) → B(q, U(2)), where the maps are induces by the double coverings SU(2) → SO(3) and S 1 × SU(2) → U(2). Acknowledgments We’d like to thank José Manuel Gómez for helpful suggestions about U(2) and for motivating us to look into SU(m) and U(m) for m > 2. 2

Non-abelian nilpotent subgroups of SU(2)

Let T ⊂ SU(2) denote the maximal torus λ 0 | λ ∈ C and |λ| = 1 T = 0 λ


4

and let w =

0 −1 . A straightforward calculation shows: 1 0

Lemma 2.1. Let x, y ∈ T . Then wx = xw and • [x, y] = 1; • [x, wy] = x2 ; • [wx, y] = y 2 ; • [wx, wy] = x2 y 2 . Definition 2.2. Let Q be a group, define inductively Γ1 (Q) = Q; Γq+1 (Q) = [Γq (Q), Q]. This is called the descending central series of Q · · · ⊂ Γq (Q) ⊂ · · · ⊂ Γ2 (Q) ⊂ Q. A group Q is nilpotent if Γq+1 (Q) = 1 for some q. The least such integer q is called the nilpotency class of Q. 2πi/n e 0 ∈ T be an n-th root of unity and let µn stand for the subgroup Let ξn = 0 e−2πi/n generated by ξn . The general quaternions Q2q+1 := µ2q ∪ wµ2q are of nilpotency class q for any q > 1. This follows from the fact that [ξ2n , w] = ξ22n = ξ2n−1 for all n ≥ 1. Lemma 2.3. Let H ⊂ T ∪ wT be a subgroup. Suppose Γr (H) = µ2n for some n > 0. Then Γr−1 (H) = µ2n+1 for all r > 2, and for r = 2 there exists t ∈ T such that tHt = Q2n+2 . Proof. We prove the case r = 2. Suppose Γ2 (H) = [H, H] = µ2n . Since the commutator generates µ2n , there exist z0 , wx0 ∈ H, with x0 ∈ T such that [z0 , wx0 ] = ξ2kn with k odd (otherwise we’d have [H, H] ⊆ µ2n−1 ). Let z ∈ H ∩ T . Then [z, wx0 ] = z 2 is a power of ξ2n , that is, z is a power of ±ξ2n+1 and z ∈ µ2n+1 . Let wy ∈ H ∩ wT , where y ∈ T . The commutator [wy, wx0] = y 2x0 2 = ξ2pn for some p > 0, therefore y = ±ξ2pn+1 x0 . Therefore H ⊆ µ2n+1 ∪ wµ2n+1 x0 . To get the equality, consider the element z0 ∈ H. We have 2 possibilities: • z0 is diagonal, and z02 = ξ2n implies z0 = ±ξ2n+1 . • z0 is anti-diagonal, so that z0 = wz0′ , with z0′ = ±ξ2n+1 x0 . In both cases z0 and wx0 z0 generate µ2n+1 ∪ wµ2n+1 x0 . Conjugating any element wξ2pn+1 x0 √ by t = x0 ∈ T we obtain √

√ √ √ x0 wξ2pn+1 x0 x0 = w x0 ξ2pn+1 x0 x0 = wξ2pn+1 .

√ This is independent of the choice of the branch cut for x0 . Hence Γ1 (tHt) = tHt ⊆ µ2n+1 ∪ wµ2n+1 . For r > 2 the same arguments without the anti-diagonal matrices cases prove the result, since Γr−1 (H) ⊂ T . Lemma 2.4. Let X, Y ∈ SU(2).


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1. If [X, Y ] = I and g ∈ SU(2) diagonalizes X, then it also diagonalizes Y . 2. If [X, Y ] = −I and g ∈ SU(2) diagonalizes X, then gXg −1 = ±ξ4 and gY g −1 ∈ wT . Proof. 2. Let g be a matrix that conjugates X to a diagonal matrix with eigenvalues λ, λ. Let X ′ = gXg −1 and Y ′ = gY g −1. Choose a non-zero vector v ∈ Eλ , the eigenspace of X associated to λ. Since [X ′ , Y ′ ] = −I we get X ′ Y ′ v = −Y ′ X ′ v = −λY ′ v. Hence −λ is an eigenvalue of X so that −λ = λ, which implies λ = ±i ∈ C. Therefore X ′ = ±ξ4 . Now, v ∈ Ei and Y ′ v ∈ E−i tells us that Y is conjugated by g into an antidiagonal matrix. Proposition 2.5. Let H ⊂ SU(2) be a nilpotent subgroup. Then either H is abelian or there exists a unique r ≥ 2 and an element g ∈ SU(2) such that gHg −1 = Q2r+1 . Proof. Suppose H is non-abelian. Then, there exists a unique r > 1 such that Γr+1 (H) = I and Γr (H) 6= I. This implies that Γr (H) sits inside the center of H. Non-abelian subgroups of SU(2) have center contained in {±I} (for a proof of this statement see [9, Example 2.22, p. 16]) and therefore Γr (H) = [H, Γr−1 (H)] = µ2 . Fix an element X in Γr−1 (H). For any h ∈ H, the commutator [X, h] is inside {±I}. By Lemma 2.4, H can be conjugated to T ∪ wT by an element g ∈ SU(2) that diagonalizes X. Now, Γr (gHg −1) = µ2 and applying Lemma 2.3 inductively we get that there exists t ∈ T , such that tgH(gt)−1 = Q2r+1 . Lemma 2.6. Let x, y ∈ T . Then • xyx−1 = y; • (wx)y(wx)−1 = y; • x(wy)x−1 = x2 (wy) = x2 yw = wx2 y; • (wx)(wy)(wx)−1 = x2 y 2 (wy) = x2 y 3 w = wx2 y 3. Lemma 2.7. 1. Let q ≥ 3. The abelian subgroups of Q2q are all subgroups of µ2q−1 or {±I, ±wx} where x is an element of µ2q−1 . 2. Let q ≥ 4. The non-abelian subgroups of Q2q are of the form µ2r−1 ∪ wxµ2r−1 where x = (ξ2q−1 )p with 0 ≤ p < 2q−r and 3 ≤ r ≤ q. Proof. 1. By Lemma 2.1, any subgroup containing an element of µ2q−1 − {±I} and one of wµ2q−1 has non-trivial commutator. 2. By the proof of Lemma 2.3 any non-abelian nilpotent subgroup of T ∪wT has the form µ2n ∪ wxµ2n for some x ∈ T . Therefore the non-abelian subgroups of Q2q are µ2r−1 ∪ wxµ2r−1 where x is not in µ2r−1 . Lemma 2.8. The normalizer of Q2q in SU(2) is: 1 + i −1 + i 1 for 1. The Binary Octahedral group which is generated by ξ8 , w, 2 1+i 1−i q = 3;


6

2. Q2q+1 for q > 3. Proof. For this it is more convenient to think of SU(2) as the group of unit quaternions, which enables one to think geometrically in terms of rotations in the space of purely imaginary quaternions, a space we shall identify with R3 . If q = cos(θ) + sin(θ)ˆ q where qˆ is a unit −1 length imaginary quaternion, then v 7→ qvq maps imaginary quaternions to imaginary quaternions and is a rotation around qˆ of angle 2θ. An arbitrary quaternion can be written as a + v where a is the real part and v is purely imaginary. We have q(a + w)q −1 = a + qvq −1, so that conjugation by q preserves the real part and rotates the imaginary part. This rule that associates a rotation of R3 to each unit quaternion is the double cover homomorphism SU(2) → SO(3). The normalizer of Q2q consists of those unit quaternions whose corresponding rotations are the orientation preserving symmetries of the set of imaginary parts of the elements of Q2q . As quaternions, the elements of Q2q are cos(2πk/2q−1)+i sin(2πk/2q−1 ) and jcos(2πk/2q−1)+ k sin(2πk/2q−1 ) for k = 0, 1, . . . , 2q−1 − 1. The imaginary parts are then 2q−2 points on the i-axis together with the vertices of a regular 2q−1 -gon in the jk-plane. For q > 3, any orientation preserving isometry of R3 preserving this set of points must separately preserve the 2q−1 -gon and the points on the i-axis, and thus must be a rotation of R3 whose restriction to the plane of the 2q−1 -gon is a symmetry of the 2q−1 -gon. Each of the 2q elements of the dihedral group of symmetries of the 2q−1 -gon extends to a unique rotation of R3 (the rotations in the dihedral group extend to rotations around the i-axis, while the reflections extend to rotations of angle π around the axis of the reflection), and each such rotation of R3 has two quaternions inducing it; these 2q+1 quaternions are easily seen to be the elements of Q2q+1 . Now, if q = 3, there is extra symmetry because the 2q−1 -gon in the jk-plane is just a square, and the 2q−2 = 2 points on the i-axis together with that square form a regular octahedron. Therefore the normalizer of Q23 is the preimage in SU(2) of the group of orientation preserving symmetries of the octahedron in SO(3);this preimage is known as 1 + i −1 + i listed in the statement of the Binary Octahedral group. The generator 21 1+i 1−i the proposition (which is the only generator not in Q24 ), corresponds to a rotation of 2π/3 around a line connecting the centers of two opposite faces of the octahedron. This extra symmetry does not separately preserve the i-axis and the square {±j, ±k}. Remark 2.9. Here’s an alternative “matrix-based” proof of the second part. Let g ∈ NSU (2) (Q2q ). Suppose gξ4g −1 = wx for some x ∈ µ2q−1 . Then gξ8g −1 is an element of order 8, and must lie in some subgroup of µ2q−1 , which is a contradiction. Therefore gξ4g −1 is a diagonal matrix and this can only be ±ξ4 . By Lemma 2.4, g is in T ∪ wT . Using Lemma 2.6, ξ2q ∈ NSU (2) (Q2q ) and for any g ∈ NSU (2) (Q2q ), g 2 is in Q2q , and thus NSU (2) (Q2q ) = Q2q+1 . Theorem 2.10. Let q ≥ 3 and n ≥ 2. Then Hom(Fn /Γqn , SU(2)) ∼ = Hom(Zn , SU(2)) ⊔ where C(n, q) =

G

P U(2)

C(n,q)

2n−2 (2n − 1)(2n−1 − 1) + (2n − 1)(2(q−3)(n−1) − 1)22n−3 . 3


7

Proof. Consider the map SU(2) × (Q2q )n → Hom(Fn /Γqn , SU(2)) given as conjugation by elements of SU(2). By Proposition 2.5 this map is surjective when we restrict to noncommuting tuples. Modding out by the center Z(SU(2)) = {±I} we have the induced surjective map ψ : P U(2) × [(Q2q )n − Hom(Zn , Q2q )] → Hom(Fn /Γqn , SU(2)) − Hom(Zn , SU(2)).

Fix an element x ∈ (Q2q )n − Hom(Zn , Q2q ) and let g, h ∈ SU(2). Then gxg −1 = hxh−1 implies that gh−1 commutes with a diagonal matrix and with an anti-diagonal matrix. Thus gh−1 = ±I. This shows that the map ψ is injective restricted to each connected component. Now, let ∼ be the equivalence relation on (Q2q )n − Hom(Zn , Q2q ) defined by: two elements are equivalent if they are conjugated to one another by some element in P U(2). We get the homeomorphism P U(2) × ([(Q2q )n − Hom(Zn , Q2q )]/ ∼) ∼ = Hom(Fn /Γq , SU(2)) − Hom(Zn , SU(2)). n

n

Let Gen(n, Q2r ) ⊂ (Q2r ) be the subset of n-tuples that generate Q2r . The normalizer NSU (2) (Q2r ) acts on Gen(n, Q2r ) by conjugation. The inclusions Gen(n, Q2r ) ⊂ (Q2r )n induce a bijective function q G Gen(n, Q2r )/NSU (2) (Q2r ) → [(Q2q )n − Hom(Zn , Q2q )]/ ∼ . r=3

The action of NSU (2) (Q2r ) on Gen(n, Q2r ) is free once we take quotient by the center of the group, which acts trivially. Thus q X |Gen(n, Q2r )| n n |[(Q2q ) − Hom(Z , Q2q )]/ ∼ | = . |NSU (2) (Q2r )|/2 r=3

By Lemma 2.8 we know the order of NSU (2) (Q2r ). It remains to count the number of elements in Gen(n, Q2r ), for which we use an inclusion–exclusion argument: a tuple generates Q2r if and only if its elements don’t all come from a single proper maximal subgroup of Q2r . So if M1 , M2 , . . . , Mm are the maximal subgroups of Q2r , we have X X X |Mi ∩ Mj ∩ Mk |n + · · · |Mi |n + |Mi ∩ Mj |n − |Gen(n, Q2r ) = |Q2r |n − i

i,j

i,j,k

For Q8 , there are three maximal subgroups, all isomorphic to Z/2 × Z/2. the intersection of any two of them is the center of Q8 , ±I. So we get |Gen(n, Q8 )| = 8n − 3 · 4n + 3 · 2n − 2n = 2n+1 (2n − 1)(2n−1 − 1).

For r ≥ 4, it follows from 2.7 that things are pretty much the same: there are just three maximal subgroups of Q2r , namely, µ2r−1 , µ2r−2 ∪ wµ2r−2 and µ2r−2 ∪ wξ22−1 µ2r−2 . And the intersection of any pair of them is µ2r−2 , so we get |Gen(n, Q2r )| = (2r )n − 3 · (2r−1 )n + 3 · (2r−2 )n − (2r−2 )n = 2(r−2)n+1 (2n − 1)(2n−1 − 1),

and a quick calculation verifies the formula claimed.

Remark 2.11. As noted in [9], the space Hom(Fn /Γ3n , SU(2)) is the same as the space of almost commuting tuples Bn (SU(2), {±I}). In [2] they describe this space, and agrees completely with our computation for q = 3.


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3

Consequences for SO(3) and U(2)

Theorem 2.10 can be used to describe the spaces of n-nilpotent tuples in SO(3) and U(2). We study first the ones in SO(3) ∼ = P U(2). Let π : SU(2) → SU(2)/{±I} ∼ = P U(2) denote the quotient homomorphism, and π∗ : Hom(Fn /Γqn , SU(2)) → Hom(Fn /Γqn , P U(2)) the induced map. We claim that for any q > 2, the image of π∗ is precisely Hom(Fn /Γnq−1 , P U(2)). Clearly any commuting tuple in SU(2) is mapped to a commuting tuple in P U(2). Now, let H be a non-abelian q-nilpotent subgroup of SU(2). As showed before this implies that Γq (H) = {±I} and hence Γq (π(H)) = I. Therefore π(H) has nilpotency class q − 1. It remains to show that any (q − 1)-nilpotent subgroup in P U(2) has a lift to a q-nilpotent subgroup of SU(2), but this follows from the fact that π is an epimorphism and the preimage of any trivial commutator is {±I}, and thus central in SU(2). Therefore, for any q ≥ 3 π∗ : Hom(Fn /Γqn , SU(2)) → Hom(Fn /Γnq−1 , P U(2))

is surjective. A result of W. M. Goldman [7, Lemma 2.2] implies that the restriction π∗−1 (C) → C to any connected component C of Hom(Fn /Γnq−1 , P U(2)), is a 2n -fold covering map (the fiber is in correspondence with Hom(Fn /Γqn , {±I}n ) = {±I}n ). From Theorem 2.10 we know that the connected components of Hom(Fn /Γqn , SU(2)) are either Hom(Zn , SU(2)) or Ki which denotes components homeomorphic to P U(2). Each of the later components consist of conjugated representations (by elements of P U(2)) of a fixed surjective homomorphism ρ : Fn /Γqn → Q2r with 3 ≤ r ≤ q. The image under π∗ of these components are the conjugated homomorphisms (also by elements of P U(2)) of the corresponding epimorphism π∗ (ρ) : Fn /Γnq−1 → D2r−1 . Since π∗ is open, π∗ | : Ki → π(Ki ) is also a covering map. Fix a representation σ : Fn /Γnq−1 → D2r−1 in π∗ (Ki ). Then the lifts of σ in Ki are gρg −1 : Fn /Γqn → gQ2r g −1 that once we mod out by {±I} we obtain ρ. Hence g is in Z(D2r−1 ). We have two different cases. First, if r ≥ 4, then Z(D2r−1 ) = hπ(ξ4)i and π∗ (Ki ) ∼ = P U(2)/π(C4 ), where C4 is the cyclic group generated by ξ4 . Therefore the restriction π∗ |Ki : Ki → π∗ (Ki ) is a 2-fold covering map. When r = 3, Z(D4 ) = D4 , and π|Ki : Ki → π∗ (Ki ) ∼ = P U(2)/π(Q8 ) is a 4-fold covering map. With these covering spaces, we can easily count the connected components of the space Hom(Fn /Γqn , SO(3)). Indeed, let Hom(Zn , SO(3))1 denote the connected component containing (I, ..., I). The connected component corresponding to Hom(Zn , SU(2)) is mapped under π∗ to Hom(Zn , SO(3))1. For components of commuting tuples, other than the comn ponent Hom(Zn , SO(3))1, the correspondence is 24 to 1, and for components of r-nilpotent n tuples with r > 2 it is 22 to 1. Thus, we only need to divide the numbers C(n, 3) and C(n, r) − C(n, 3) of Theorem 2.10 by 2n−2 and 2n−1 respectively. Corollary 3.1. Let q ≥ 2, n ≥ 2. Then

Hom(Fn /Γqn , SO(3)) ∼ = Hom(Zn , SO(3))1 ⊔ n

G

M (n,2)

S 3 /Q8 ⊔

G

S 3 /C4 ,

M (n,q)

n−1

where M(n, 2) = (2 −1)(23 −1) , M(n, q) = (2n − 1)(2(q−3)(n−1) − 1)2n−2 and C4 is the cyclic group of order 4 generated by ξ4 . Both Q8 and C4 act by conjugation on S 3 . Remark 3.2. The case q = 2 was originally computed in [8], and the number M(n, 2) agrees with their calculation.


9

situation for U(2). Any matrix in X ∈ U(2) can be written as p Now we′ discuss the det(X)X , where X ′ ∈ SU(2). In this decomposition, [X, Y ] = [X ′ , Y ′ ] for any X, Y in U(2). Consider the map (S 1 )n × Hom(Fn /Γqn , SU(2)) → Hom(Fn /Γqn , U(2)) given by (λ1 , . . . , λn , x1 , . . . , xn ) 7→ (λ1 x1 , . . . , λn xn ). By the previous observation this a surjective map. These representations are uniquely determined up to a negative sign, that is, we have a homeomorphism (S 1 )n ×(Z/2)n Hom(Fn /Γqn , SU(2)) ∼ = Hom(Fn /Γqn , U(2)). Using Theorem 2.10 we get that the connected components of this space are homeomorphic to (S 1 )n ×(Z/2)n Hom(Zn , SU(2)) ∼ = Hom(Zn , U(2)) or (S 1 )n ×H P U(2) where H is a subgroup n of (Z/2) . The first component corresponds to the commuting n-tuples in U(2) and the later to the r-nilpotent tuples with 2 ≤ r ≤ q. We can see the (Z/2)n action as an action on the indexing set of the connected components, which can be represented by elements of Hom(Fn /Γqn , Q2q ). To count the number of connected components, let ~ε = (ε1 , . . . , εn ) with each εi = ±1 be an arbitrary element in (Z/2)n and ~x = (x1 , . . . , xn ) a non-commutative n-tuple in Hom(Fn /Γqn , Q2q ). Then the stabilizer of this element consists of Stab(~x) = {~ε | (ε1 x1 , . . . , εn xn ) = (gx1 g −1, . . . , gxn g −1) for some g ∈ SU(2)}. That is, such g either commutes or anticommutes with all xi . We have several cases. Let ~ε ∈ Stab(~x) be a non trivial element. Case 1: Suppose some xi lies in the torus T . • If εi = 1, xi = gxi g −1 and thus g must also lie in T . To generate an r-nilpotent subgroup of Q2q with xi , we need at least one more element of the form ξ2kq w for some k. This element does not commute with any element of T − {±I} and only anticommutes with ±ξ4 . Thus, the only choices for g are ±I and ±ξ4 . • If εi = −1, −xi = gxi g −1 and by Lemma 2.4, xi = ±ξ4 and g ∈ T w. Again, in the n-tuple there must be an element of the form xj = ξ2kq w for some k. The only elements in T w that commute or anticommute with xj are g = ±ξ2kq w or g = ±ξ4 ξ2kq w. Note that in this case, the remaining elements in the n-tuple can only be of the form ±I, ±ξ4 , ±ξ2kq w or ±ξ4 ξ2kq w, which generates a copy of Q8 . Case 2: Suppose all xi lie in T w. Let xi = ξ2kq w. • Suppose ~x is an r-nilpotent tuple with r ≥ 3. Then there is at least one xj = ξ2l q w that is different from ±xi or ±ξ4 xi . Thus the only choices for g are ±I and ±ξ4 . • Suppose ~x is a 2-nilpotent tuple. Then the only other choices for the remaining xj ’s are ±ξ2kq w or ±ξ4 ξ2kq w. Hence g can only be ±I, ±ξ4 , ±ξ2kq w or ±ξ4 ξ2kq w. We can conclude that if ~x generates a copy of Q8 , then |Stab(~x)| = 4 and |Stab(~x)| = 2 in any other case.

10


Corollary 3.3. Let q ≥ 3 and n ≥ 2, then G G (S 1 )n ×(Z/2) P U(2), (S 1 )n ×(Z/2)2 P U(2) ⊔ Hom(Fn /Γqn , U(2)) ∼ = Hom(Zn , U(2)) ⊔ f(n,3) M

f M(n,q)

f(n, q) = M(n, q − 1) is as in Corollary 3.1. where M

Remark 3.4. It was proved in [4, Theorem 1] that given a finitely generated nilpotent group Γ, a complex reductive linear group G and K ⊂ G a maximal compact subgroup, the inclusion induces a strong deformation retract Hom(Γ, G) ≃ Hom(Γ, K). Applying this to G = SL(2, C), SL(3, R) or GL(2, C) and K = SU(2), SO(3) or U(2) respectively, we also get a homotopy type description of Hom(Fn /Γqn , G) for all q ≥ 2. 4

2-nilpotent tuples in U(m)

What about m > 2, what are the connected components of Hom(Fn /Γqn , U(m)) then? We can not give an answer to this in its wide generality, but we can at least say something about the case q = 3. Let a be a partition of {1, 2, . . . , m} into disjoint non-empty subsets. Define U(a) as the subgroup of U(m) consisting of m × m “block diagonal matrices with blocks indexed by a”, by which we mean matrices A ∈ U(m) whose (i, j)-th entry is 0 whenever i and j are in different parts of the partition a. To explain our terminology, notice that when each part of a consists of consecutive numbers, say, if the parts are {1, . . . , m1 }, {m1 + 1, . . . , m1 + m2 }, {m1 + m2 + 1, . . . , m1 + m2 + m3 }, . . ., then A is what is traditionally called a block diagonal matrix:   A1 0   .. A= Ai ∈ U(mi ) ; . 0 Ak The conjugacy class of the subgroup U(a) depends only on the sizes of the parts of a. To be specific, if π is any permutation of {1, . . . , m} such that the image of each part of a consists of consecutive numbers, then U(a) is conjugate, via the permutation matrix associated to π, to the subgroup of traditional block diagonal matrices as above (where the m Qik are the sizes of the parts of a). In particular, the subgroup U(a) is always isomorphic to i=1 U(mi ) where the mi are the sizes of the parts of a but the isomorphism is not at all unique. Let Za denote the center of U(a) which consists of “block scalar matrices”: diagonal matrices diag(λ1 , . . . , λm ) ∈ U(m) such that λi = λj whenever i and j are in the same part of a. For example, if the parts consist of consecutive numbers, the elements of Za are of the form:   0 λ1 Im1   ..  . . 0 λk Imk

Given any diagonal matrix D = diag(λ1 , . . . , λm ) ∈ U(m) there is a coarsest partition a(D) such that D ∈ Za(D) , namely, the partition where i and j are in the same part if and only if λi = λj . One can easily check that the centralizer of D is precisely U(a(D)).


11

Our goal is to interpret 2-nilpotent tuples of U(m) as almost commuting elements of the the subgroups U(a), that is, for any topological group G and any closed subgroup K ⊂ G contained in the center of G, the space of K-almost commuting n-tuples is the subspace of Gn where each (x1 , . . . , xn ) satisfies that [xi , xj ] lies in K for all i, j. We denote this space as Bn (G, K) and we have an inclusion Bn (G, Z(G)) ⊂ Hom(Fn /Γ3n , G). In particular, for U(m), [ Bn (U(a), Za ) ⊂ Hom(Fn /Γ3n , U(m)), a⊢m

where we’ve borrowed the notation a ⊢ m typically used for partitions of the number m to indicate that the union is over all partitions of the set {1, . . . , m} where each parts consists of consecutive numbers and the parts are ordered by size. Let Rep(Γ, G) denote the orbit space of the action of G on Hom(Γ, G) by conjugation. Proposition 4.1. The above inclusion is surjective upon passing to orbits, that is, ! [ Bn (U(a), Za ) /U(m) = Rep(Fn /Γ3n , U(m)). a⊢m

Remark 4.2. The notation −/U(m) on the left is not meant suggest the union is closed under the conjugation action! It just means the image of the union in the orbit space. Proof. Let (x1 , . . . , xn ) be an element of Hom(Fn /Γ3n , U(m)). Then every commutator [xi , xj ] is central in the group generated by {x1 , . . . , xn }. In particular, all commutators commute with each other. Since each xi is in U(m) and hence diagonalizable, we can simultaneously diagonalize all commutators by an element g ∈ U(m). Let yi := gxi g −1 and yij := g[xi , xj ]g −1 . Now, each yij lies in the center Zaij for some coarsest partition aij . Choose a as the infimum of all the aij , that is, as the coarsest partition refining all aij . We have by construction yij ∈TU(aij ) ⊆ U(a); and for each k, we have that yk is in the centralizer of each yij , so yk ∈ k U(aij ) = U(a). The last remaining detail is that this partition a may not have parts that consist of consecutive numbers, or those parts may not be ordered by size, but, as explained above, a further conjugation fixes that. Remark 4.3. The same argument works for SU(m) and its subgroups SU(a). There are only two Q minor differences: the first is that SU(a) is identified with the subgroup of matrices in ki=1 U(mi ) of determinant 1; the second is that for the last bit of the proof, the “consecutivization”, one needs to observe that for every partition one can always find an even permutation such that the image of each part consists of consecutive numbers and permutation matrices for even permutations lie in SU(m). 5

Stable Homotopy type

In our previous description of Hom(Fn /Γqn , SU(2)), one of the connected components is Hom(Zn , SU(2)). In order to give a characterization of the homotopy type of the space of


12

nilpotent n-tuples we need one of the commuting n-tuples in SU(2). Unfortunately this has not been yet obtained. Nevertheless, the recognition of the stable homotopy type has been achieved independently in [2], [3] and [5]. Let S1 (Fn /Γqn , SU(2)) denote the subspace of Hom(Fn /Γqn , SU(2)) consisting of n-tuples with at least one coordinate equal to I. In [2] they show that Hom(Z , SU(2))/S1(Z , SU(2)) ∼ = n

n

S3 if n = 1 2 nλ2 2 (RP ) /sn (RP ) if n ≥ 2

(1)

where (RP)nλ2 is the associated Thom space of nλ2 , n times the Whitney sum of the universal bundle λ2 over RP2 , and sn is its zero section. Using the homotopy stable decomposition of the simplicial space induced by the commuting tuples in a closed subgroup of GLn (C) (proved by the same authors in [1]) they get a complete description of Hom(Zn , SU(2)) after one suspension. In [3], they prove that 

 ΣHom(Zn , SU(2)) ≃ Σ 

n _

k=1

n (_ k)



 ΣS(kλ2 )

where S(kλ2 ) is the sphere bundle associated to kλ2 . These two decompositions agree since ΣS(kλ2 ) ≃ (RP2 )kλ2 /sn (RP2 ). In M. C. Crabb’s paper [5], he expresses the stable homotopy type of Hom(Zn , SU(2))+ as a wedge of various copies of RP2 , RP4 /RP2 and RP5 /RP2 . Now, it was proved in [9] that if G is a compact Lie group, then there are G-equivariant homotopy equivalences   n ( k) _  _ ΣHom(Fn /Γqn , G) ≃ Σ  Hom(Fk /Γqk , G)/S1 (Fk /Γqk , G) 1≤k≤n

for all n and q. As a consequence of this and Theorem 2.10, we get the following: Corollary 5.1. Let n ≥ 1 and q ≥ 3. There are homotopy equivalences    n ) ( k _ _ _  _ ΣHom(Fn /Γqn , SU(2)) ≃ Σ S 3 Σ  (RP2 )kλ2 /sk (RP2 ) ∨ RP3+  n

2≤k≤n

K(k,q)

where q

K(n, q) =

X (2r − 1)n − 3(2r−1 − 1)n + 2(2r−2 − 1)n 7n 3n 1 − + + . 24 8 12 r=4 2r

(2)


13

Proof. For each q ≥ 3, let H1,q (SU(2)) denote the complement of Hom(Zn , SU(2)) in Hom(Fn /Γqn , SU(2)) and similarly, let S1,q (SU(2)) denote the complement of S1 (Zn , SU(2)) in S1 (Fn /Γqn , SU(2)). We want to describe Hom(Fn /Γqn , SU(2))/S1 (Fn /Γqn , SU(2)), which is homeomorphic to Hom(Zn , SU(2))/S1 (Zn , SU(2)) ∨ H1,q (SU(2))/S1,q (SU(2)). The right hand wedge sumand can be identified with the one point compactification (H1,q (SU(2)) − S1,q (SU(2)))+ .

F Recall that in the proof of Theorem 2.10, H1,q (SU(2)) is shown to be C(n,q) P U(2) by separating the non-abelian nilpotent n-tuples ~x according to the value of r for which the subgroup generated by ~x is conjugate to Q2r . Since S1,q (SU(2)) consists of non-commuting ntuples with at least one entry equal to the identity I, we can use similar reasoning but now for generating tuples F without any coordinate equal to I, to obtain that the space H1,q (SU(2)) − S1,q (SU(2)) ≃ K(n,q) P U(2), where (23 − 1)n − 3(22 − 1)n + 2(2 − 1)n 7n 3n 1 = − + , 24 24 8 12 F + and K(n, q) for q ≥ 4 is as in the statement of the Corollary. Finally, P U(2) = K(n,q) W K(n,q) P U(2)+ . The remaining quotient spaces in (2) arise from commuting tuples, and these are given by (1). The result now follows. K(n, 3) =

In [2] they also show that Hom(Z , SO(3))1/S1 (Z , SO(3))1 ∼ = n

n

RP3 if n = 1 2 nλ2 (RP ) if n ≥ 2

(3)

We use this to compute the homotopy stable decomposition of Hom(Fn /Γqn , SO(3)). Corollary 5.2. Let n ≥ 1 and q ≥ 2. Then ΣHom(Fn /Γqn , SO(3)) is homotopy equivalent to    n ) ( k _ _ _ _  _ (S 3 /Q8 )+ ∨ (S 3 /C4 )+  Σ  (RP2 )kλ2 ∨ Σ RP3 n

2≤k≤n

N (k)

N (k,q)

where

q

X (2r − 1)n − 3(2r−1 − 1)n + 2(2r−2 − 1)n 1 N(n) = (3n−1 − 1) and N(n, q) = 2 2n+r−2 r=3 q Proof. Consider the map π∗ : Hom(Fn /Γq+1 n , SU(2)) → Hom(Fn /Γn , SO(3)) as in the proof q+1 of Corollary 3.1. Then π∗−1 (S1 (Fn /Γqn , SO(3))) = S1 (Fn /Γq+1 n , SU(2)) ∪ S−1 (Fn /Γn , SU(2)) where the subscript −1 corresponds to the n-tuples with at least one coordinate equal to −I.


14

By [7, Lemma 2.2], the restrictions π∗ |−1 (C) → C are covering maps for every connected component C, where π∗ | is the restriction to −1 q q q Hom(Fn /Γq+1 n , SU(2))−π∗ (S1 (Fn /Γn , SO(3))) → Hom(Fn /Γn , SO(3))−S1 (Fn /Γn , SO(3)).

Now we count the number of connected components of −1 q E := Hom(Fn /Γq+1 n , SU(2)) − π∗ (S1 (Fn /Γn , SO(3)))

which are the (q +1)-nilpotent n-tuples in SU(2) with no entries equal to I or −I. As before, the number of connected components of non-commuting nilpotent n-tuples of this subspace can be counted as (23 − 2)n − 3(22 − 2)n + 2(2 − 2)n 2n (3n−1 − 1) = , for q = 3, and 24 8 q X (2r − 2)n − 3(2r−1 − 2)n + 2(2r−2 − 2)n r=4

2r

for q ≥ 4.

Recall that the correspondence of components of the total space E to the ones in the base space is 2n−2 to 1 for components of commuting tuples different of Hom(Zn , SU(2)) and 2n−1 to 1 for q > 2. Dividing the above numbers by 2n−2 and 2n−1 respectively, we get the number of connected components of Hom(Fn /Γqn , SO(3)) − S1 (Fn /Γqn , SO(3)). The one point compactification of this space yields all the nilpotent stable pieces in (2) different from Hom(Zn , SO(3))1/(Hom(Zn , SO(3))1 ∩ S1 (Zn , SO(3))), which is given by (3). The result now follows from Corollary 3.1. Remark 5.3. For q = 3, the number of wedges homeomorphic to RP3+ in Corollary 5.1 agrees with the description made in [2] for ΣBn (SU(2), {±I}). Also, the stable decomposition of Hom(Zn , SO(3)) was first computed in [2] and the number of wedges homeomorphic to (S 3 /Q3 )+ in Corollary 5.2 agrees with N(n). Now we describe the spaces of representations Rep(Fn /Γqn , G) of our groups of interest. In [2] they show that Rep(Zn , SU(2))/((S1 (Zn , SU(2))/SU(2)) ∼ = S n /Σ2 where the action of the generating element in Σ2 is given by (x0 , x1 , . . . , xn ) 7→ (x0 , −x1 , . . . , −xn ) for any (x0 , . . . , xn ) in S n . Identifying S n with the suspension ΣS n−1 , we can see the above action as first taking antipodes on S n−1 and then suspending, that is S n /Σ2 = ΣRPn−1 . Therefore    n ( k) _ _  _ S 0 ∨ ΣRPk−1  . Σ  ΣRep(Fn /Γqn , SU(2)) ≃ 1≤k≤n

K(k,q)

Similarly

  n (_ k) _   q Σ ΣRep(Fn /Γn , SO(3)) ≃ 1≤k≤n

_

N (k)+N (k,q)

  S 0 ∨ ΣRPk−1  .


15

For G = U(2), we use [2, Theorem 6.1, p. 473] to get that Rep(Zn , U(2))/(S1 (Zn , U(2))/U(2)) ∼ = (S 1 × S 1 )∧n /Σ2 ,

where X ∧n denotes the smash product of n copies of X and Σ2 acts by simultaneously swapping the S 1 factors in each smash factor (note that for a single smash factor, S 1 ×S 1 /Σ2 is a Möbius band). Then, for q > 2,    n ( k) _ _ _  Σ  S 0 ∨ (S 1 × S 1 )∧k /Σ2  . ΣRep(Fn /Γqn , U(2)) ≃ 1≤k≤n

6

N (k)+N (k,q+1)

Cohomology of B(r, Q2q ) for r = 2 and 3

Now we turn our attention to the classifying spaces of transitionally q-nilpotent G-bundles for a topological group G. As described in [1], the spaces Hom(Fn /Γqn , G) ⊂ Gn give rise to a simplicial subspace of the nerve of G. The geometric realizations B(q, G) := |Hom(F∗ /Γq∗ , G)| fit into a natural filtration of BG Bcom G := B(2, G) ⊂ B(3, G) ⊂ · · · ⊂ B(q, G) ⊂ · · · ⊂ BG. So far we have completely described the subgroups of Q2q . This will allow us to compute the homotopy type of B(r, Q2q ) as follows. Let G be a finite group. Consider the category Pr (G) with set of objects {Mα , Mα ∩ Mβ } where Mα are the maximal subgroups of G of nilpotency class at most r. The set of arrows in Pr (G) is given by identities and inclusions. It was proved in [1] that when Pr (G) is a tree, there is a homotopy equivalence B(r, G) ≃ B colim A . A∈Nr (G)

Let q ≥ 4. By Lemma 2.7 we conclude for 2 < r < q q−r 2q−2 2 r Z/4 ∗ Q B(r, Q2q ) ≃ B ∗Z/2 r−1 2 Z/2 ∗Z/2

where ∗ denotes the amalgamated product of groups.

2 Z/4)) Cohomology of Bcom Q2q : Taking r = 2 we get that Bcom Q2q ≃ B(Z/2q−1 ∗Z/2 (∗Z/2 for any q ≥ 3. Applying the associated Mayer-Vietoris sequence inductively we obtain  Z n=0  q−2 H n (Bcom Q2q ; Z) ∼ n even = Z/2q−1 ⊕ (Z/2)2  0 otherwise. q−2

Proposition 6.1. H ∗ (Bcom Q8 ; F2 ) ∼ = F2 [y1 , y2 , y3, z]/(yi yj , y12 + y22 + y32 , i 6= j) where yi has degree 1 and z degree 2. Let q ≥ 4. Then the F2 -cohomology ring of Bcom Q2q is given by H (Bcom Q2q ; F2 ) ∼ = ∗

F2 [x1 , x2 , y1 , . . . , y2q−2 , z]/(x21 , xk yi , yi yj , i

where x1 , yj have degree 1 and x2 , z have degree 2.

6= j, x2 +

q−2 2 X

i=1

yi2),

16


2 Proof. We work out the case q ≥ 4. Let Γ = Z/2q−1 ∗Z/2 (∗Z/2 Z/4). We use the central extension q−2 Z/2 ⊳ Γ → Z/2q−2 ∗ ∗2 Z/2. Recall that the cohomology rings for n > 1, H ∗ (Z/2n ; F2 ) ∼ = F2 [x1 , x2 ]/x21 where deg(x1 ) = 1 ∗ q−2 2q−2 and deg(x2 ) = 2. Thus, H (Z/2 ∗∗ Z/2; F2 ) ∼ = F2 [x1 , x2 , y1 , . . . , y2q−2 ]/(x21 , xk yi , yi yj , i 6= P2q−2 2 yi . j). The k-invariant of the associated Serre spectral sequence of this extension is x2 + i=1 Therefore the E3∗,∗ page is q−2

F2 [z] ⊗ F2 [x1 , x2 , y1 , . . . , y2q−2 ]/(x21 , xk yi , yi yj , i 6= j, x2 +

q−2 2 X

yi2 ).

i=1

P2q−2 2 yi ) = Sq 1 (x2 ) is 0 since it can be expressed only in The Steenrod square Sq 1 (x2 + i=1 terms of x2 . That is, d3 = 0 and thus the spectral sequence abuts to E3∗,∗ . So we have found the E∞ -page along with its ring structure. Since all the relations involve only generators from the base of the fibration, these relations hold in the cohomology ring as well. Cohomology of B(3, Q16 ): We know that B(3, Q16 ) ∼ = B(∗2Z/4 Q8 ∗Z/2 ∗4Z/2 Z/4). To compute the F2 cohomology of this space, first we analyze G1 = Q8 ∗Z/4 Q8 by looking at the central extension Z/2 ⊳ G1 → Z/2 × (Z/2 ∗ Z/2) and its k-invariant. Recall that H ∗ (Z/2 × (Z/2 ∗ Z/2); F2 ) ∼ = F2 [x1 , x2 , z]/(x1 x2 ) with all generators of degree 1. By naturality, the k-invariant of this extension restricts to Z/2 _

Z/2 _

Q8

l

/

Z/2 × Z/2

G1

/

Z/2 × (Z/2 ∗ Z/2)

where l stands for the inclusion of one of the copies of Q8 in G1 . The k-invariant of the left extension in the diagram is x2 + xy + y 2, where x, y are the degree 1 generators in H ∗ ((Z/2)2 ; F2 ) (recall that the cohomology ring H ∗ (Q8 ; F2 ) = F2 [x, y, t]/(x2 + xy + y 2 , x2 y + xy 2 ) where t has degree 4). Since the k-invariant of the right extension restricts to the inclusion of both copies of Q8 in G1 we conclude that the k-invariant of the desired extension must be x21 + x22 + z 2 + x1 z + x2 z. Now, let G2 = ∗4Z/2 Z/4. The central extension Z/2 ⊳ G2 → ∗4 Z/2 has k-invariant y12 + y22 + y32 + y42 where H ∗ (∗4 Z/2; F2 ) ∼ = F2 [y1 , y2, y3 , y4 ]/(yi yj , i 6= j) with all yi of degree 1. Using these k-invariants we can compute the cohomology of G1 ∗Z/2 G2 . Consider the central extension Z/2 ⊳ G1 ∗Z/2 G2 → (Z/2 × (Z/2 ∗ Z/2)) ∗ ∗4 Z/2.


17

Let A denote the cohomology ring H ∗ (Z/2 × (Z/2 ∗ Z/2) ∗ ∗4 Z/2; F2 ) which using the same notation as before, is isomorphic to F2 [x1 , x2 , y1 , y2 , y3, y4 , z]/(x1 x2 , yi yj , xk yi , zyi i 6= j). Let l denote 1 or 2. With a similar argument as for G1 , the commutative diagram Z/2 _

Z/2 _

l

Gl

/ G1

∗Z/2 G2

/ Z/2 × (Z/2 ∗ Z/2) ∗

Gl /Z/2

∗4Z/2

implies that the k-invariant for the right central extension in the diagram must be k = x21 + x22 + z 2 + x1 z + x2 z + y12 + y22 + y32 + y42 . Using the Kudo and Serre transgression theorems, the differential d2 in the E2 -page of the Lyndon-Hochschild-Serre spectral sequence maps the generator t in H ∗ (Z/2; F2 ) = F2 [t] to k. We claim is that k is not a zero divisor in A; a claim which we verified using the computer algebra system Singular as explained in the appendix. Assuming k is not a zero divisor we can calculate the E3 -page as follows: P i ∗,∗ ∼ Writing an element in the E -page, E F [t] ⊗ A, as a = ai t , where ai ∈ A, we = 2 2 2 P i−1 2 have that d2 (a) = iai t k. We see that ker(d2 ) = A[t ], im(d2 ) = kA[t2 ], so we get ∗,∗ ∼ 2 E3 = F2 [t ] ⊗ A/(k) where t2 has degree 2. Next, d3 (t2 ) = Sq 1 (k) = x21 z + x1 z 2 + x22 z + x2 z 2 and its image in the E3 -page is z 3 . Again, writing a in the E3 page as ai ∈ A/(k), we can conclude that ker d3 /im d3 which is the E4 -page is

P

ai (t2 )i , with

E4∗,∗ ∼ = A/(k, z 3 )[t4 ] ⊕ t2 annA/(k) (z 3 )[t4 ]. where we claim the annihilator annA/(k) (z 3 ) is given by (y1 , y2, y3 , y4 ); again we used Singular to verify this, as explained in the appendix. The remaining differentials that could be non-zero are given by n

n−1

d2n +1 (t2 ) = Sq 2

n−1

Sq 2

n

n

n

. . . Sq 2 Sq 1 (k) = (x21 + x22 )z + (x1 + x2 )z 2

∗,∗ for n ≥ 2. We’ll now show that all of these are in fact already zero, so that E∞ = E4∗,∗ . n n On the E4 -page z 3 = 0, so the differentials are just (x21 + x22 )z, and these are all divisible by the first one, (x41 + x42 )z, so it is enough to show that one vanishes. This folows from the following easily verified identities in A:

(x31 + x32 )zk = (x41 + x42 )z + (x21 + x22 )z 3 + (x31 + x32 )z 2 (x1 + x2 )z 2 k = (x31 + x32 )z 2 + (x1 + x2 )z 4 + (x21 + x22 )z 3 . The above formula for E4∗,∗ describes the ring structure of the E∞ -page; we are not claiming to have shown that this ring structure in the E∞ -page is also the ring structure of H ∗ (B(3, Q16 )).


18

7

Homotopy type of B(q, SU(2))

We finish with a homotopy pushout description of the spaces B(q, SU(2)). Consider the map P U(2) × (Q2q )n → Hom(Fn /Γqn , SU(2)) where (g, x1 , . . . , xn ) 7→ (gx1 g −1, . . . , gxn g −1 ) and q > 2. It is well defined in the quotient P U(2) ×Nq (Q2q )n → Hom(Fn /Γqn , SU(2)) where the normalizer Nq := NP U (2) (D2q−1 ) acts by translation on P U(2) and by conjugation on (Q2q )n . Consider the subsets Gen(n, Q2q ), Hom(Fn /Γnq−1 , Q2q ) ⊂ (Q2q )n and the restrictions of the above map to the respective subspaces ∅

q−1 P U(2) ×Nq Hom(F n /Γn , Q2q )

/

_

P U(2) ×Nq Gen(n, Q2q )

/

Hom(Fn /Γnq−1 , SU(2))

_

/

P U(2) ×Nq (Q2q )n /

Hom(Fn /Γqn , SU(2)).

We claim that all the above squares are pushouts. From the proof of Theorem 2.10 we have the homeomorphism P U(2) × Gen(n, Q2q )/Nq ∼ = Hom(Fn /Γqn , SU(2)) − Hom(Fn /Γnq−1 , SU(2)). Since Gen(n, Q2q ) is discrete, Nq is finite and acts freely, we have that P U(2) × Gen(n, Q2q )/Nq ∼ = P U(2) ×Nq Gen(n, Q2q ) which proves the outside square to be a pushout in sets. But, Hom(Fn /Γnq−1 , SU(2)) is closed and P U(2) ×Nq Gen(n, Q2q ) is the image of a compact space, so that Hom(Fn /Γqn , SU(2)) has the disjoint union topology. This proves our claim for the outside square, and a similar argument can be used for the square on the left side. Now we look at the right square. The outside and left squares being pushouts imply the right square is also a pushout. The middle arrow is a closed cofibration, and thus the right square is also a homotopy pushout. Moreover, since the maps are either inclusions or given by conjugation, all arrows are simplicial maps. Geometric realization commutes with colimits and homotopy colimits, and hence P U(2) ×Nq B(q − 1, Q2q ) /

B(q − 1, SU(2))

P U(2) ×Nq BQ2q /

B(q, SU(2))

is a pushout of topological spaces and a homotopy pushout. Finally, here are some relations with the classifying spaces B(q, SO(3)) and B(q, U(2)). Let q ≥ 3 and consider the map πn := π∗ : Hom(Fn /Γqn , SU(2)) → Hom(Fn /Γnq−1 , SO(3)). In section 3 we said that πn is a 2n -fold covering map for every n ≥ 0. Moreover, we have the pullback diagrams Hom(Fn /Γqn , SU(2))

/

SU(2)n and Hom(Zn , SU(2)) (π)n

πn

Hom(Fn /Γnq−1 , SO(3))

/

SO(3)n

/

SU(2)n (π)n

πn

Hom(Zn , SO(3))1

/

SO(3)n .


19

Again, all maps in the square are simplicial. Since geometric realization of simplicial spaces commutes with taking pullbacks, we obtain pullback squares of topological spaces: B(q, SU(2))

BSU(2) and Bcom SU(2) /

B(q − 1, SO(3)) /

/ BSU(2)

BSO(3)

Bcom SO(3)1

/

BSO(3).

The right hand side arrow is a BZ/2-bundle, hence the two arrows on the left are also principal BZ/2-bundles. Now, let q ≥ 2 and consider the group homomorphism ψ : S 1 × SU(2) → U(2) given by (λ, x) 7→ λx, where S 1 represents the scalar matrices in U(2). The kernel of ψ is the group of order 2 generated by (−1, −I). For each n we have the pullback square (S 1 )n × Hom(Fn /Γqn , SU(2))

/

(S 1 × SU(2))n (ψ)n

πn

Hom(Fn /Γqn , U(2))

/

U(2)n

which for the same reasons as before, after taking geometric realizations gives a BZ/2-bundle BZ/2 → BS 1 × B(q, SU(2)) → B(q, U(2)). A

Computing annihilators in Singular

We made two claims about annihilators in the above calculation: that annA (k) = 0 and annA/(k) (z 3 ) = (y1 , y2 , y3, y4 ). We verified these claims with help from the Singular computer algebra system [6]. We include the code used in order for the benefit of potential readers facing similar computations but unfamiliar with what computer algebra systems can offer and how easy to use they can be. ring P = 2, (x1,x2,y1,y2,y3,y4,z), dp; ideal I = x1*x2, y1*y2, y1*y3, y1*y4, y2*y3, y2*y4, y3*y4, x1*y1, x1*y2, x1*y3, x1*y4, x2*y1, x2*y2, x2*y3, x2*y4, z*y1, z*y2, z*y3, z*y4; qring A = std(I); poly k = x1^2 + x2^2 + x1*z + x2*z + z^2 + y1^2 + y2^2 + y3^3 + y4^2; quotient(0, k); This defines P as the polynomial ring F2 [x1 , x2 , y1 , y2 , y3 , y4, z]; the 2 specifies the coefficients are from F2 . Then it defines A as the quotient P/I (when A is defined, P is the “current ring” and thus the quotient is implicitly a quotient of A), and asks for generators of the ideal (0 : (k)). Singular responds _[1]=0, indicating that k is not a zero divisor. To compute the annihilator of z 3 in A/(k), we can do: qring A_mod_k = std(k); quotient(0, z^3);

20


Singular responds with generators for the annihilator: _[1]=y4, _[2]=y3, _[3]=y2, _[4]=y1, _[5]=x1^2+x2^2+x1*z+x2*z+z^2 Notice the last one is actually equal to y12 + y22 + y32 + y42 in A/(k), so the annihilator is as we claimed. We also needed to show that (x41 + x42 )z is zero in A/(k). We can check this by asking for reduce(x1^4*z+x2^4*z, std(z^3)); when A_mod_k is the current ring. References [1] A. Adem, F. Cohen, E. Torres-Giese. “Commuting elements, simplicial spaces and filtrations of classifying spaces”, Math. Proc. Cambridge Philos. Soc. Vol. 152 (2012), 1, 91-114. [2] A. Adem, F. Cohen, J. M. G´ omez. “Stable splittings, spaces of representations and almost commuting elements in Lie groups”, Math. Proc. Camb. Phil. Soc. Vol. 149 (2010), 455-490. [3] T. Baird, L. Jeffrey, P. Selick. “The space of commuting n-tuples in SU (2)”, Illinois Journal of Mathematics, Volume 55, Number 3, Fall 2011, Pages 805-813. [4] M. Bergeron. “The Topology of Nilpotent Representations in Reductive Groups and their Maximal Compact Subgroups”, Geometry and Topology 19-3 (2015), 1383–1407. [5] M. C. Crabb. “Spaces of commuting elements in SU (2)”, Proceedings of the Edinburgh Mathematical Society (2011) 54, 67-75. [6] W. Decker, G.-M. Greuel, G. Pfister, H. Sch¨ onemann. Singular 4-1-0 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2016). [7] W. M. Goldman “Topological components of spaces of representations”, Invent. Math. 93 (1988), 557–607. [8] E. Torres-Giese, D. Sjerve “Fundamental groups of commuting elements in Lie groups”, Bull. London Math. Soc. (2008) 40 (1) 65-76. [9] B. Villarreal. “Cosimplicial arXiv:1601.04688.

groups

and

spaces

of

homomorphisms”,

submitted,

Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2, Canada E-mail adress: [email protected] Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2, Canada E-mail adress: [email protected]