COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS

ALEJANDRO ADEM^{∗} AND FREDERICK R. COHEN^{∗∗}

Abstract. This article records basic topological, as well as homological properties of the space of homomorphisms Hom(π, G) where π is a finitely generated discrete group, and G is a Lie group, possibly non-compact. If π is a free abelian group of rank equal to n, then Hom(π, G) is the space of ordered n–tuples of commuting elements in G. If G = SU(2), a complete calculation of the cohomology of these spaces is given for n = 2, 3. An explicit stable splitting of these spaces is also obtained, as a special case of a more general splitting.

1. Introduction

Let π denote a finitely generated discrete group and G a Lie group. The set of homomorphisms Hom(π, G) from π to G can be topologized in a natural way as a subspace of a finite product of copies of G, where the relations on π carve out what can be a variety with interesting singularities. The purpose of this paper is to describe the geometry and cohomology of these spaces for certain classes of groups.

These spaces are interesting and can be quite complicated. The approach in this article is to start by studying their basic structure. Additional methods for analyzing Hom(π, G) are developed by restricting to suitable classes of discrete groups π, and varying the target group G.

First consider the question of path–connectivity for these spaces. In the case of G = O(n) or G = SO(n), the first two Stiefel–Whitney classes play an important role:

Theorem 1.1. For a finitely generated discrete group π, the following two statements hold:

• There is a decomposition into non–empty, disjoint closed subsets (called w–sectors):

Hom(π, O(n)) ≅ ⨆_{w∈H^{1}(π,𝔽_{2})}Hom(π, O(n))_{w}

Date: April 13, 2007.

∗Partially supported by the NSF and NSERC.

∗∗Partially supported by the NSF, grant number 0340575.

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 2

where a w–sector corresponds to those homomorphisms with first Stiefel–Whitney class equal to w. In particular if H^{1}(π, 𝔽_{2}) ≠ 0, then Hom(π, O(n)) is not path– connected. The sector for w = 0 is precisely Hom(π, SO(n)).

• For n sufficiently large, Hom(π, O(n))_{w} has at least as many components as the cardinality of the image of H^{2}(π/[π, π], 𝔽_{2}) → H^{2}(π, 𝔽_{2}).

This applies very well to fundamental groups of certain complements of complex hyperplane arrangements, such as the pure braid groups:

Corollary 1.2. If π is the fundamental group of the complement of an arrangement of complex hyperplanes which has contractible universal cover, and if n is sufficiently large, then

#π_{0}(Hom(π, O(n)) ≥ |H^{1}(π, 𝔽_{2})||H^{2}(π, 𝔽_{2})|

Kapovich and Millson have investigated the geometry of Hom(π, G) near the identity for π equal to the fundamental group of complements of certain arrangements [11], Theorem 1.13. The approach in this article is to give cruder, more global information such as the number of path-components.

The number of path-components may change when the target is restricted from a non– compact Lie group G to a maximal compact subgroup K. Examples are given in §3 where the natural map Hom(π, K) → Hom(π, G) fails to be surjective on path-components. Classically, these arise in the case where π is the fundamental group of a closed orientable surface of genus at least two with G = SL(2, ℝ) and K = SO(2). A further example given here arises from π = Γ_{2} the mapping class group of a surface of genus two, where U(2) ⊂ Sp_{4}(ℝ) is a maximal compact subgroup. In this case, Hom(Γ_{2}, U(2)) → Hom(Γ_{2}, Sp_{4}(ℝ)) fails to be surjective on path components (see 3.2).

Under certain conditions, the space of homomorphisms must be path-connected. This is the case when π = ℤ^{n} and G is a Lie group with path–connected maximal abelian subgroups. For certain classes of groups this can then be used to analyze the natural map Hom(π, G) → [Bπ, BG], where [Bπ, BG] denotes the based homotopy classes of maps between the classifying spaces Bπ and BG. The following theorem is proved in Section 2:

Theorem 1.3. Let π denote a homologically toroidal group^{1}; then, if G is a Lie group with path–connected maximal abelian subgroups, the map

B_{0} : π_{0}(Hom(π, G)) → [Bπ, BG]

1A group π is said to be homologically toroidal if there exists a finite free product H of free abelian groups of finite rank and a homomorphism H → π which induces a split epimorphism in integral homology.

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 3

is trivial. In particular, if ℳ is the complement of an arrangement of complex hyperplanes which is aspherical, then any complex unitary representation of π = π_{1}(ℳ) induces a trivial bundle over Bπ.

The second part of this paper is specialized to the cohomology of the space of ordered commuting n–tuples of elements in a compact Lie group G. The main results here mostly apply to the case when G = SU(2). The approach is to consider the complement, namely the space of ordered non–commuting n–tuples; which seems to be much more tractable. Duality is then used to compute the cohomology of the commuting n–tuples. For example, the complement of Hom(ℤ^{2}, SU(2)) in SU(2) × SU(2) is homotopy equivalent to SO(3), and the cohomology of the commuting pairs can be easily computed from this. The case of commuting triples in SU(2)^{3} requires more intricate calculations, which yield the following results:

Theorem 1.4.

H^{i}(Hom(ℤ ⊕ ℤ, SU(2)), ℤ) ≅ \{ \begin{matrix}ℤ if & i = 0 \\ 0 if & i = 1 \\ ℤ if & i = 2 \\ ℤ ⊕ ℤ if & i = 3 \\ ℤ/2ℤ if & i = 4 \\ 0 if & i ≥ 5 \end{matrix}

H^{i}(Hom(ℤ ⊕ ℤ ⊕ ℤ, SU(2)), ℤ) ≅ \{ \begin{matrix}ℤ if & i = 0 \\ 0 if & i = 1 \\ ℤ ⊕ ℤ ⊕ ℤ if & i = 2 \\ ℤ ⊕ ℤ ⊕ ℤ if & i = 3 \\ ℤ/2ℤ ⊕ ℤ/2ℤ ⊕ ℤ/2ℤ if & i = 4 \\ ℤ if & i = 5 \\ 0 if & i = 6 \\ ℤ/2ℤ if & i = 7 \\ 0 if & i ≥ 8 \end{matrix}

In the last part of this paper, a stable splitting is given for the space of commuting n–tuples. Natural subspaces of Hom(ℤ^{n}, G) arise from the so-called fat wedge filtration of the product G^{n} where the base-point of G is 1_{G}. Thus if F_{j}G^{n} is the subspace of G^{n} with at least j coordinates equal to 1_{G}, define subspaces of Hom(ℤ^{n}, G) by the formula S_{n}(j, G) = Hom(ℤ^{n}, G) ∩ F_{j}G^{n}, and write S_{n}(G) = S_{n}(1, G).

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 4

A Lie group G is said have cofibrantly commuting elements if the natural inclusions I_{j} : S_{n}(j, G) → S_{n}(j − 1, G) are cofibrations for all n and j for which both spaces are non–empty. Many Lie groups satisfy this last property.

Theorem 1.5. If G is a closed subgroup of GL(n, ℂ), then G has cofibrantly commuting elements.

In what follows, if X and Y are pointed topological spaces, then X ⋁ Y denotes their one point union in the product X × Y and ΣX denotes the suspension of X.

Theorem 1.6. If G is a Lie group which has cofibrantly commuting elements, then there are homotopy equivalences

Σ(Hom(ℤ^{n}, G)) ≃ ⋁_{1≤k≤n} Σ(\begin{matrix}( & {n \atop k} & ) \\ ⋁ \end{matrix} Hom(ℤ^{k}, G)/S_{k}(G)).

The last two theorems imply the next corollary.

Corollary 1.7. If G is a closed subgroup of GL(n, ℂ), then there are isomorphisms of graded abelian groups in cohomology (which may not preserve products)

H^{∗}(Hom(ℤ^{n}, G), ℤ) → H^{∗}( ⋁_{1≤k≤n} \begin{matrix}( & {n \atop k} & ) \\ ⋁ \end{matrix} Hom(ℤ^{k}, G)/S_{k}(G), ℤ).

Applying this to the case of ordered commuting pairs in SU(2), there is a homotopy equivalence which holds after a single suspension:

Hom(ℤ^{2}, SU(2)) ≃ SU(2) ⋁ SU(2) ⋁(𝕊^{6} − SO(3))

where (𝕊^{6} −SO(3)) is the Spanier–Whitehead dual of SO(3) in 𝕊^{6}. These methods also yield the result that after one suspension, Hom(ℤ^{3}, SU(2)) is homotopy equivalent to

⋁^{3} SU(2) ⋁[⋁^{3}(𝕊^{6} − SO(3))] ⋁[SU(2) ∧ (𝕊^{6} − SO(3)].

A general decomposition for Hom(ℤ^{n}, SU(2)) is described, but the summands have not yet been identified as concretely in terms of Spanier-Whitehead duals.

The authors would like to acknowledge the hospitality and support of the Max-Planck-Institut für Mathematik in Bonn, the Pacific Institute for the Mathematical Sciences (Vancouver) and the Institute for Advanced Study (Princeton). They would like to thank P. Selick and L. Jeffrey for pointing out a mistake in the homology calculation in an earlier

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 5

version of this paper; Enrique Torres for his careful reading of a preliminary version of this manuscript; and Bill Browder, Zinovy Reichstein, Karl H. Hofmann and D. Z. Djokovic for conversations concerning Lie groups.

2. General Properties of Hom(π, G)

Basic general properties of spaces of homomorphisms Hom(π, G) are given in this section. A key reference here is the paper by W.Goldman [7], although the approach will be much less geometric.

Definition 2.1. Let π denote a finitely generated group, and G a finite dimensional Lie group; Hom(π, G) will denote the set of all group homomorphisms from π to G.

Let F_{n} denote the free group with n generators {x_{1}, x_{2}, ⋯, x_{n}}. There is an evaluation map E : Hom(F_{n}, G) → G^{n} defined by E(f) = (f(x_{1}), f(x_{2}), ⋯, f(x_{n})). There is a second evaluation map e : G^{n} → Hom(F_{n}, G) defined by e(g_{1}, g_{2}, ⋯, g_{n}) = f where f(x_{i}) = g_{i}. The composites e ∘ E, and E ∘ e are both the identity, and thus these maps are invertible.

Note that if Γ → π is a surjection, it induces an inclusion Hom(π, G) → Hom(Γ, G). In particular, if π is a finitely generated group, let F_{n} → π denote a fixed surjection; endow the space of homomorphisms with the natural subspace topology Hom(π, G) ⊂ G^{n} or in other words make the inclusion map a homeomorphism onto its image in the Cartesian product. Given two surjections of this type, F_{n} → π and F_{m} → π, there is a surjection F_{n+m} → π which factors through both of the original surjections, and hence gives that the topology on Hom(π, G) does not depend on the choice of generators. More generally, a surjection of finitely generated discrete groups Γ → π induces a homeomorphism of Hom(π, G) onto its image in Hom(Γ, G). Hom(π, G) can be regarded as a pointed space, for which the base-point is the constant map f : π → G with f(x) = 1_{G}, the identity element in G.

Observe that G acts by conjugation on the space of homomorphisms; the associated orbit space Rep(π, G) = Hom(π, G)/G is often called the representation space. Note that the projection onto this quotient is highly singular, and can have a complicated structure.

The space Hom(π, G) can in fact be realized as the fixed point set of a π action on the pointed mapping space [12]. Consider Map_{∗}(π, G), the space of all maps from π to G which map 1 to 1. Then there is an action of π on this space defined as follows; for f : π → G, g, h ∈ π, define

gf(h) = f(hg)f(g)^{−1}

from which it follows that

Hom(π, G) = Map_{∗}(π, G)^{π}

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 6

If π is a finite group, then the space of homomorphisms is the fixed point set of a smooth action on a compact manifold (namely G^{|π|−1}), hence is itself a smooth manifold, not necessarily connected.

The following proposition records the facts above as well as some straightforward consequences.

Proposition 2.2. Let π denote a finitely generated discrete group, and G a Lie group.

(1) If Γ and π are finitely generated discrete groups with p : Γ → π a surjective homomorphism, then the induced map

p^{∗} : Hom(π, G) → Hom(Γ, G)

is a homeomorphism onto its image. In particular if Γ = F_{n}, and π is discrete, then p^{∗} : Hom(π, G) → Hom(F_{n}, G) is a homeomorphism onto its image in G^{n}. Note that in this case, Hom(π, G) is Hausdorff.

(2) If

1 → G → H → K → 1

is a group extension, then

1 → Hom(π, G) → Hom(π, H) → Hom(π, K)

is an exact sequence of sets where the base-point in Hom(π, G) is the constant map c_{G} with c_{G}(x) = 1_{G} for all x in π.

(3) If π is isomorphic to the free product of discrete groups A ∗ B, then there is a home-

omorphism

Hom(π, G) ≅ Hom(A, G) × Hom(B, G).

(4) If G is an abelian group, then the natural map

Hom(π/[π, π], G) → Hom(π, G)

is a homeomorphism. Hence if π/[π, π] ≅ F ⊕ T where F is free abelian and T is torsion, then

Hom(π, 𝕊^{1}) ≅ (𝕊^{1})^{rk F} × T.

(5) If π is a finite group, then Hom(π, G) is a smooth submanifold of G^{|π|−1}.

Important examples arise when considering ordered n–tuples of commuting elements in a Lie group G; they can be identified with Hom(ℤ^{n}, G). A key question is whether or not these spaces are path–connected.^{2} Proposition 2.3. If every abelian subgroup of G is contained in a path-connected abelian subgroup, then the space Hom(ℤ^{n}, G) is path-connected.

2Note that if G itself is abelian, then there is an equivalence Hom(ℤ^{n}, G) ≅ G^{n}.

Proof. Assume that every abelian subgroup of G is contained in a path-connected abelian subgroup. Let e_{1}, …, e_{n} denote the standard basis for ℤ^{n}; given any homomorphism f : ℤ^{n} → G, consider the images of these generators: x_{1} = f(x_{1}), …, x_{n} = f(e_{n}). These elements commute, hence x_{1}, x_{2}, …, x_{n} all lie in a path-connected abelian group 𝕋. Choose paths p_{i} : [0, 1] → 𝕋 ⊂ G from p_{i}(0) = 1 to p_{i}(1) = x_{i} for all i = 1, …, n. Define a homotopy H : [0, 1] → Hom(ℤ^{n}, G) by the formula H(t)(e_{i}) = p_{i}(t); by multiplying, this extends to a well–defined homotopy in the space of homomorphisms between the trivial homomorphism and our given homomorphism f. □

Corollary 2.4. If G is equal to one of the compact groups U(n), SU(n) or Sp(n); or equal to GL_{n}(ℂ), then Hom(ℤ^{n}, G) is path–connected.

Proof. For the compact groups listed above, the maximal abelian subgroups are precisely the maximal tori. For the case of GL_{n}(ℂ) this follows from the fact that the centralizer of every subset is connected (see [15], 3.22). □

Remark. The conclusion of the corollary is not satisfied for n sufficiently large with G = SO(n) or Spin(n).

Let Map_{∗}(Bπ, BG) denote the space of based maps between the classifying spaces of π and G respectively. Its set of path–components is denoted by [Bπ, BG] and corresponds to based homotopy classes of maps between the two classifying spaces. The following result is well–known.

Lemma 2.5. The natural map B : Hom(π, G) → Map_{∗}(Bπ, BG) given by f ↦ Bf is continuous, and there is an induced map of sets

B_{0} : π_{0}(Hom(π, G)) → [Bπ, BG].

These maps factor through the quotient Rep(π, G) = Hom(π, G)/G, and the natural map π_{0}(Hom(π, G)) → π_{0}(Rep(π, G)) is a bijection of sets is G is path–connected.

Note that the classifying space functor is continuous as long as the compactly generated topology is used on the source, which is the case here. Clearly it makes sense to use B_{0} to distinguish path–components in the space of homomorphisms. More generally the following basic question arises:

Problem: Give conditions on π and G which imply that B_{0} : π_{0}(Hom(π, G)) → [Bπ, BG] is a bijection.

The motivation of course is to understand the contribution of representation theory to principal G–bundles over Bπ. In special circumstances e.g. when Bπ is a compact manifold, this is geometrically significant.

Example 2.6. Consider the case when π = π_{1}(M_{g}), and M_{g} is a compact orientable Riemann surface of genus g > 1. By [13], [9], if G is a compact, connected, semisimple Lie group, there is a bijection π_{0}(Hom(π, G)) ≅ H^{2}(M_{g}, π_{1}(G)). Note that in this example, M_{g} = Bπ; given a homomorphism f : π → G, there is an induced map Bf : M_{g} → BG. The bijection is obtained by looking at the map induced in homology H_{2}(M_{g}, ℤ) → H_{2}(BG, ℤ) ≅ π_{2}(BG) ≅ π_{1}(G); note that it factors through the map B_{0}. Moreover, using the standard cofibration

⋁_{i=1}^{2g} 𝕊^{1} → M_{g} → 𝕊^{2},

it follows that there is a bijection [M_{g}, BG] ≅ π_{1}(G), whence the map B_{0} must also be a bijection. In contrast, for the case of genus g = 1, recall that Hom(ℤ^{2}, U(n)) is path connected, even though π_{1}(U(n)) ≠ 1. Hence in this case the map B_{0} is trivial and so fails to be bijective.

There is a class of groups built out of free abelian groups, and which are geometrically quite significant. The definition below appears in [1]:

Definition 2.7. A discrete group π is said to be homologically toroidal if there exists H, a finite free product of free abelian groups of finite rank as well as a homomorphism ϕ : H → π which induces a split epimorphism in integral homology.

The standard examples of such groups are fundamental groups associated to arrangements of hyper-planes. More precisely, let 𝒜 denote a finite arrangement of affine hyper-planes in ℂ^{n}, and let ℳ ≅ ℂ^{n} \ ∪_{H∈𝒜} H denote their complement. Then, if this complement has contractible universal cover, π = π_{1}(ℳ) is a homologically toroidal group. The best known examples here are the pure braid groups P_{n}.

The next result was stated in the introduction as Theorem 1.3.

Theorem 2.8. Let π denote a homologically toroidal group; then, if G is a Lie group with path–connected maximal abelian subgroups, the map

B_{0} : π_{0}(Hom(π, G)) → [Bπ, BG]

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 9

is trivial. In particular any complex unitary representation of such groups induces a trivial vector bundle over Bπ.

Proof. There is a commutative diagram:

π_{0}(Hom(π, G)) ⟶^{B_{0}} [Bπ, BG]
↓^{ϕ^{∗}}  ↓^{Bϕ^{∗}}
π_{0}(Hom(H, G)) ⟶^{B′_{0}} [BH, BG]

Note that as H ≅ H_{1} ∗ ⋯ ∗ H_{r}, where each H_{i} is free abelian of finite rank, then Hom(H, G) ≅ Hom(H_{1}, G)×⋯×Hom(H_{r}, G); this is a product of path connected spaces– hence it is also path–connected. Therefore the composition Bϕ^{∗} · B_{0} is trivial. However, as was shown in [1], page 529, the inverse image of every point under Bϕ^{∗} consists of a single element, as this is just the Puppe exact sequence induced by the cofibration BH → Bπ → C_{Bϕ} where the second map is null-homotopic by our hypotheses. Hence it follows that B_{0} is trivial, verifying the claim. □

Corollary 2.9. If ℳ is the complement of an arrangement of hyperplanes which is aspherical, then every unitary representation of π_{1}(ℳ) induces a trivial bundle over ℳ.

Next consider target groups with maximal abelian subgroups which are not connected. The basic examples are the orthogonal groups O(n), the special orthogonal groups SO(n) and their double covers, the spinor groups Spin(n).

If O(n) is the usual real orthogonal group, then there is a split extension

1 → SO(n) → O(n) → ℤ/2 → 1.

The double cover of SO(n) is called Spin(n); it fits into a central group extension

1 → ℤ/2 → Spin(n) → SO(n) → 1.

When n = 3, there is the classical extension

1 → ℤ/2 → SU(2) → SO(3) → 1.

Hence there are exact sequences of pointed sets

1 → Hom(π, SO(n)) → Hom(π, O(n)) → Hom(π, ℤ/2)

and

1 → Hom(π, ℤ/2) → Hom(π, Spin(n))) → Hom(π, SO(n)).

Note that the term Hom(π, ℤ/2) is equal to H^{1}(π, ℤ/2) (mod 2 group cohomology).

As observed earlier, O(n) is a semidirect product O(n) ≅ SO(n) ×_{T} ℤ_{2} (in fact for n odd this is a direct product). Using a section ℤ/2 → O(n) for the determinant map, the following can be established:

Lemma 2.10. The determinant map

ψ : Hom(π, O(n)) → H^{1}(π, 𝔽_{2})

is a split surjection, with each

ψ^{−1}(w) = Hom(π, O(n))_{w}

a non–empty open (and closed) subspace. This decomposes Hom(π, O(n)) into disjoint subspaces:

Hom(π, O(n)) ≅ ⨆_{w∈H^{1}(π,𝔽_{2})}Hom(π, O(n))_{w}

and in particular

#π_{0}(Hom(π, O(n)) = ∑_{w∈H^{1}(π,𝔽_{2})}#π_{0}(Hom(π, O(n))_{w}).

Corollary 2.11. If H^{1}(π, 𝔽_{2}) ≠ 0, then Hom(π, O(n)) is not path–connected.

The subspace Hom(π, O(n))_{w} ⊂ Hom(π, O(n)) will be referred to as the w–sector of the space of homomorphisms.

Consider the Stiefel–Whitney classes w_{i}(f) = f^{∗}(w_{i}) ∈ H^{i}(π, 𝔽_{2}) associated to an orthogonal representation f : π → O(n). These are homotopy invariants of Bf : Bπ → BO(n), hence they are well defined functions [Bπ, BO(n)] → H^{i}(π, 𝔽_{2}) which can, in some situations, detect path components for the space of homomorphisms. Note that f : π → O(n) lands in SO(n) if and only if w_{1}(f) = 0, and further it will lift to Spin(n) if and only if w_{2}(f) = 0. The subspace Hom(π, O(n))_{w} consists of all homomorphisms f : π → O(n) with first Stiefel–Whitney class equal to w, and the 0–sector is precisely Hom(π, SO(n)).

The next result makes use of the second Stiefel–Whitney class.

Lemma 2.12. Let A be a finitely generated abelian group and w a fixed class in H^{1}(π, 𝔽_{2}). Then for n sufficiently large, the natural map

Hom(A, O(n))_{w} → H^{2}(A, 𝔽_{2})

given by f ↦ w_{2}(f) is surjective.

Proof. The proof of this for w = 0 (i.e. Hom(A, SO(n))) is routine, and follows easily from analogues of the arguments given in [1], pp. 532–533. Now for general w ≠ 0 observe that there is a one–dimensional representation ϵ : A → O(1) with first Stiefel–Whitney class equal to w. Given v ∈ H^{2}(A, 𝔽_{2}), realize it as w_{2}(f) for some f : A → SO(m). Now let f′ denote the composition i · (f × ϵ), where i is the natural inclusion SO(m) × O(1) → O(m + 1) given by

(A, t) ↦ (\begin{matrix}A & 0 \\ 0 & t \end{matrix}) .

A simple calculation shows that w_{1}(f′) = w_{1}(ϵ) + w_{1}(f) = w whereas w_{2}(f′) = w_{2}(f) + w_{1}(f)w_{1}(ϵ) + w_{2}(ϵ) = v. □

This lemma can used to prove the following result.

Theorem 2.13. Let π denote a finitely generated discrete group; then for n sufficiently large and w ∈ H^{1}(π, 𝔽_{2}),

#π_{0}(Hom(π, O(n))_{w}) ≥ N(π)

where N(π) = |im[H^{2}(π/[π, π], 𝔽_{2}) → H^{2}(π, 𝔽_{2})]|.

Proof. Consider the commutative diagram associated to the abelianization map

ρ : π → π/[π, π]
π_{0}(Hom(π/[π, π], O(n))_{w}) ⟶^{w_{2}} H^{2}(π/[π, π], 𝔽_{2})
↓^{ρ^{∗}}  ↓^{Bρ^{∗}}
π_{0}(Hom(π, O(n))_{w}) ⟶^{w_{2}} H^{2}(π, 𝔽_{2})

By the previous lemma, the top arrow is a surjection for n sufficiently large. It follows that the bottom arrow surjects onto the image of Bρ^{∗} : H^{2}(π/[π, π], 𝔽_{2}) → H^{2}(π, 𝔽_{2}). □

Corollary 2.14. Let π be a finitely generated discrete group such that

H^{2}(π/[π, π], 𝔽_{2}) → H^{2}(π, 𝔽_{2})

is surjective; then for n sufficiently large,

#π_{0}(Hom(π, O(n))) ≥ |H^{1}(π, 𝔽_{2})||H^{2}(π, 𝔽_{2})|.

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 12

These results apply particularly well to fundamental groups of aspherical complements of hyperplane arrangements. Their cohomology is torsion free and generated by 1–dimensional classes, hence in fact for these groups their cohomology is a quotient of the cohomology of their abelianization.

Proposition 2.15. Let ℳ denote the complement of an arrangement of hyperplanes which happens to be aspherical; if π denotes its fundamental group and n is sufficiently large, then

#π_{0}(Hom(π, O(n)) ≥ |H^{1}(π, 𝔽_{2})||H^{2}(π, 𝔽_{2})|.

So far, the contribution of the first two Stiefel–Whitney classes to the number of pathcomponents of Hom(π, O(n)) has been analyzed. However, this is very far from being the only situation where the space of homomorphisms fails to be path-connected. Indeed it has been shown in [10] that Hom(ℤ^{3}, Spin(7)) is not path-connected, even though (using methods similar to those outlined before, see [1]) the map

π_{0}(Hom(ℤ^{n}, Spin(m))) → [Bℤ^{n}, BSpin(m)]

is trivial, for all m, n ≥ 1.

3. Examples

One basic fact about spaces of homomorphisms is that they can change rather drastically even when the target groups are closely related. In particular if K ⊂ G is a maximal compact subgroup, the set of path-components may change when considering Hom(π, K) → Hom(π, G). This will be illustrated with examples where π is allowed to be more general than a finitely generated free abelian group.

Let G = SL(2, ℝ) with maximal compact subgroup given by SO(2). Note that SO(2) is abelian while SL(2, ℝ) contains a free group on two letters (even as a discrete subgroup). Thus, there is a homeomorphism

Hom(H_{1}(π), SO(2)) ≅ Hom(π, SO(2)),

but the natural map

Hom(H_{1}(π), SL(2, ℝ)) → Hom(π, SL(2, ℝ))

is not a surjection on path-components in the case when π is the fundamental group of a Riemann surface of genus greater than one; notice that there are faithful representations of surface groups in SL(2, ℝ) by classical work of Fricke and Klein [6] (this also follows from [13]).

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 13

Now let Γ_{g} denote the mapping class group of a closed orientable surface of genus g, and let B_{n} denote Artin’s braid group on n strands. Consider homomorphisms into the group P GL(2, ℂ), which has maximal compact subgroup given by SO(3),

The following result has classical origins, but does not appear to be standard.

Proposition 3.1. Assume that π is the mapping class group Γ_{g} for g ≥ 2, or B_{n} for n ≥ 6.

(1) If G = SO(3), any homomorphism ρ : π → G factors through a cyclic group, and thus the natural map

Hom(H_{1}(π), SO(3)) → Hom(π, SO(3))

is a homeomorphism.

(2) The space Hom(B_{n}, SO(3)) is homeomorphic to SO(3) for n > 5.

(3) The space Hom(Γ_{g}, SO(3)) is a point for g > 2, and is Hom(ℤ/20ℤ, SO(3)) in case g = 2.

Proof. Recall that the first homology group of the braid group B_{n} is ℤ for n > 1 while the first homology group of the mapping class group Γ_{g} is trivial for g > 2 with H_{1}(Γ_{2}) = ℤ/20ℤ. Thus, Hom(H_{1}(π), SO(3)) is either SO(3) in case π = B_{n}, a point in case π = Γ_{g} for g > 2, or Hom(ℤ/20ℤ, SO(3)) a closed subspace of SO(3) in case π = Γ_{2}. The abelianization map induces an embedding onto a closed subspace of Hom(π, SO(3)). To prove part (1), it suffices to see that this map is surjective.

First assume that π is B_{n} for n ≥ 6. The braid groups have the following classical presentation:

• σ_{i} · σ_{j} = σ_{j} · σ_{i} if |i − j| > 2 with 1 ≤ i, j ≤ n − 1 and

• σ_{i} · σ_{i+1} · σ_{i} = σ_{i+1} · σ_{i} · σ_{i+1} with 1 ≤ i < i + 1 ≤ n − 1.

Thus all generators are conjugate. Hence if ρ : π → SO(3) is a homomorphism, all elements ρ(σ_{i}) are rotations through equal angles θ, with axis of rotation given by 𝕏_{i}.

Assume that θ ≠ π, 0. Since σ_{1} commutes with σ_{i} for i > 2, it follows that ρ(σ_{i}) is in the maximal torus containing ρ(σ_{1}), 𝕋_{1}. Thus the axis of rotation 𝕏_{1} is equal to 𝕏_{i} for i > 2, the rotations are equal, and ρ(σ_{1}) = ρ(σ_{i}) for i > 2. A similar argument gives that ρ(σ_{2}) = ρ(σ_{i}) for i > 3. Hence if θ ≠ π, 0, then ρ(σ_{1}) = ρ(σ_{i}) for i > 1, and the first two statements follow.

It remains to check the cases of θ = 0, and θ = π with the first case being trivial. If θ = π, then assume that ρ(σ_{1}) is rotation through the z-axis, and is given by the matrix

(\begin{matrix}−1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix}) .

Since ρ(σ_{1}) commutes with ρ(σ_{i}) for i > 2, ρ(σ_{i}) is equal to

(\begin{matrix}a(i) & b(i) & 0 \\ c(i) & d(i) & 0 \\ 0 & 0 & e(i) \end{matrix}) .

Since each of these matrices is in SO(3), and the square of these matrices are the identity as θ is rotation through π, e(i)^{2} = 1 = a(i)^{2} = d(i)^{2} with c(i) = 0 = b(i). Thus these matrices are diagonal with entries ±1, the image ρ(σ_{i}) for i ≠ 2 lies in an abelian group, the 2-torus (ℤ/2ℤ)^{2}. By the relation σ_{i} · σ_{i+1} · σ_{i} = σ_{i+1} · σ_{i} · σ_{i+1}, it follows that ρ(σ_{1}) = ρ(σ_{i}) for i ≥ 3.

A similar argument using the fact that σ_{i} commutes with σ_{2} gives that ρ(σ_{2}) = ρ(σ_{i}) for i > 3. Thus ρ(σ_{1}) = ρ(σ_{j}) for j ≥ 2 and the natural map

Hom(H_{1}(π), SO(3)) → Hom(π, SO(3))

is a continuous bijection which must be a homeomorphism as both spaces are compact and Hausdorff.

A similar argument works for Γ_{g} with g ≥ 2. Namely, the mapping class group Γ_{g} is generated by σ_{i} for 1 ≤ i ≤ 2g + 2 plus one more element δ which commutes with σ_{j} for j ≠ 4 while all the generators are conjugate (by B. Wajnryb’s presentation, see [17]). Parts (2) and (3) follow from the above. □

Next, consider Sp(4, ℝ) with maximal compact subgroup equal to U(2).

Theorem 3.2. The natural map

Hom(Γ_{2}, U(2)) → Hom(Γ_{2}, Sp(4, ℝ))

fails to be a surjection on path-components.

Proof. The method of proof is to use the results in 3.1 together with properties of the map

B : Hom(π, G) → Map_{∗}(Bπ, BG)

which sends a homomorphism g to Bg together with the induced map on the level of cohomology.

Consider the composite

Γ_{2} ⟶^{f} Sp(4, ℤ) ⟶^{i} Sp(4, ℝ)

where i : Sp(4, ℤ) → Sp(4, ℝ) is the natural inclusion. Assume that this composite lifts up to conjugacy to the maximal compact subgroup U(2) of Sp(4, ℝ) to give a commutative

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 15

diagram

Γ_{2} ⟶^{i∘f} U(2)
↓^{1}  ↓
Γ_{2} ⟶^{i∘f} Sp(4, ℝ).

There is an induced commutative diagram

Γ_{2} ⟶^{i∘f} U(2) ⟶^{π} SO(3)
↓^{1}  ↓^{i}  ↓
Γ_{2} ⟶^{i∘f} Sp(4, ℝ) ⟶^{π} P Sp(4, ℝ)

By Proposition 3.1, any map

Γ_{2} ⟶ SO(3)

factors through a cyclic group. Thus the image in mod-2 cohomology in any dimension is of rank at most one.

By the results in [4] restricting to a semi-dihedral subgroup of order sixteen, SD_{16}, and the Klein 4-group in Γ_{2}, the composite

BΓ_{2} → BSp(4, ℝ) → BP Sp(4, ℝ)

satisfies the property that w_{2}^{3}, and w_{3}^{2} are linearly independent in H^{6}(BΓ_{2}; 𝔽_{2}). Thus there does not exist a representation Γ_{2} → U(2) which is homotopic to i ∘ f in Hom(Γ_{2}, Sp(4, ℝ)). The theorem follows. □

4. Some Homological Computations

Let A denote a discrete group, and G a Lie group. Consider the space Hom(ℤ × A, G). Using the surjection ℤ∗A → ℤ×A, the space Hom(ℤ×A, G) can be regarded as a subspace given by

Hom(ℤ × A, G) ⊂ G × Hom(A, G).

Proposition 4.1. Let G denote a non–abelian Lie group and A a finitely generated discrete group such that A/[A, A] ≠ {1}. Then the natural projection map pr_{1} : G×Hom(A, G) → G restricts to a map

π : G × Hom(A, G) − Hom(ℤ × A, G) → G

with the following properties:

• the image of π is G − Z(G), where Z(G) ⊂ G is the center of G

• if x_{0} ∈ G − Z(G), then
π^{−1}(x_{0}) ≅ Hom(A, G) − Hom(A, Z_{G}(x_{0}))
where Z_{G}(x_{0}) is the centralizer of x_{0} in G.

Proof. Note that if x_{0} ∈ Z(G), then for any f ∈ Hom(A, G), imf ⊂ Z_{G}(x_{0}), hence x_{0} ∉ imπ. Now if x_{0} ∉ Z(G), then for all m ≥ 0 there exists an element u_{m} ∈ G of order m, such that x_{0} does not commute with u_{m}. By the hypotheses, there exists an m > 0 and a surjection A → ℤ/m, hence a homomorphism A → G which factors through the subgroup generated by u_{m}, such that the image of f is not contained in Z_{G}(x_{0}). Hence the pair (x_{0}, f) defines an element in the domain of π, and so the first assertion has been verified.

For the second part, note that if x_{0} ∈ G − Z(G), then

π^{−1}(x_{0}) = {(x_{0}, f) | imf ⊄ Z_{G}(x_{0})}

which is clearly homeomorphic to the complement Hom(A, G) − Hom(A, Z_{G}(x_{0})). □

This proposition can be applied to an important example.

Example 4.2. Let A = F_{r}, the free group on r generators. In this case there is a projection

π : G^{r+1} − Hom(ℤ × F_{r}, G) → G − Z(G)

with π^{−1}(x_{0}) ≅ G^{r} − [Z_{G}(x_{0})]^{r}.

In general this map can be quite complicated; the different strata of the base, corresponding to elements with different centralizers, requires some rather intricate gluing to understand the homology of the domain. However by choosing G = SU(2), the situation simplifies. The main point is that there are exactly two kinds of elements in SU(2): the singular ones, which are central (namely Z(SU(2)) = {±I}) and the regular elements x_{0}, which have a maximal torus as centralizer, i.e. Z_{G}(x_{0}) = 𝕊^{1}. Under these conditions, the map π is locally trivial.

Proposition 4.3. For A as above, π is a locally trivial bundle

π : SU(2) × Hom(A, SU(2)) − Hom(ℤ × A, SU(2)) → SU(2) − {±I}

with fiber F ≅ Hom(A, SU(2)) − Hom(A, 𝕊^{1}).

Note that the base of this bundle has the homotopy type of 𝕊^{2}.

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 17

Example 4.4. Let F_{r} denote the free group on r generators. Then

π : SU(2)^{r+1} − Hom(ℤ × F_{r}, SU(2)) → SU(2) − {±I}

is a locally trivial bundle with fiber SU(2)^{r} − 𝕋^{r}, where 𝕋^{r} is the torus of rank r. Taking r = 1 yields a fibration up to homotopy

SU(2) − 𝕊^{1} → SU(2) × SU(2) − Hom(ℤ ⊕ ℤ, G) → 𝕊^{2}

for which the total space is homotopy equivalent to SO(3).

A geometric argument is required to justify this last statement as well as to supply details necessary for homology calculations. Consider the commutator map

∂ : SU(2) × SU(2) → SU(2).

Given the homeomorphism Hom(ℤ ∗ ℤ, SU(2)) ≅ SU(2) × SU(2), Hom(ℤ ⊕ ℤ, SU(2)) can be identified with the subspace ∂^{−1}(I). Let 𝒮 ⊂ Hom(ℤ ∗ ℤ, SU(2)) denote the reducible representations. These are those pairs of representations which can be simultaneously conjugated into the maximal torus, in this case the diagonal matrices. Consider the following basic fact from Akbulut-McCarthy, [2] III.2.1, pp. 53–54:

Proposition 4.5. ∂ is a surjective map, and the set of critical points for ∂ is precisely equal to 𝒮.

Now note that in fact 𝒮 = ∂^{−1}(I), as any pair of commuting matrices in SU(2) can be simultaneously diagonalized. This is exactly the subspace of singular points and hence it follows that

Proposition 4.6. M = SU(2) × SU(2) \ 𝒮 is an open smooth 6–dimensional manifold.

Next observe that SO(3) = SU(2)/{±I} acts via conjugation on M, and as observed by Akbulut-McCarthy, this action is free, and the orbit space M/SO(3) is a 3–dimensional manifold. The next result in [2], (VI.1.1) determines the homotopy type of this complement.

Proposition 4.7. Let R_{−} = ∂^{−1}(−I), then

∂_{|} : SU(2) × SU(2) \ Hom(ℤ ⊕ ℤ, SU(2)) → SU(2) \ {I}

is a locally trivial bundle, with fiber R_{−}.

Note that for general position reasons, the total space must be connected. Hence it follows that R_{−} is connected. Now note that the free SO(3) action restricts to a free action on R_{−}, with orbit space a single point. In other words R_{−} ≅ SO(3) (see [2], pp 154).

Corollary 4.8. There is a homotopy equivalence

SU(2) × SU(2) \ Hom(ℤ ⊕ ℤ, SU(2)) ≃ SO(3).

The next ingredient is crucial for subsequent calculations:

Lemma 4.9. The map in homology induced by the inclusion

i_{∗} : H_{3}(R_{−}, ℤ) → H_{3}(SU(2) × SU(2), ℤ)

is zero.

Proof. As in [2], R_{−} = {(A, B) ∈ SU(2) × SU(2) | ABA^{−1}B^{−1} = −I}. This condition implies that tr(A) = tr(B) = 0. Now, identify

𝕊^{2} = {A ∈ SU(2) | tr(A) = 0},

R_{−} ⊂ 𝕊^{2} × 𝕊^{2} ⊂ 𝕊U(2) × SU(2)

and the result follows. □

The space of homomorphisms Hom(ℤ ⊕ ℤ, SU(2)) is an analytic variety (see [7], page 568) hence locally contractible. Therefore Poincaré–Lefschetz Duality (see [5], pp. 292) can be applied to the pair (SU(2) × SU(2), Hom(ℤ ⊕ ℤ, SU(2))), to conclude that there is an isomorphism for all i ≥ 0:

H^{i}(Hom(ℤ⊕ℤ, SU(2)), ℤ) ≅ H_{6−i}(SU(2)×SU(2), SU(2)×SU(2)\Hom(ℤ⊕ℤ, SU(2)), ℤ).

The complement can be identified up to homotopy with R_{−} ≃ SO(3). From the lemma above and the long exact sequence in homology for the pair (SU(2) × SU(2), R_{−}), it follows that

H^{i}(Hom(ℤ ⊕ ℤ, SU(2)), ℤ) ≅ \{ \begin{matrix}ℤ if & i = 0 \\ 0 if & i = 1 \\ ℤ if & i = 2 \\ ℤ ⊕ ℤ if & i = 3 \\ ℤ/2ℤ if & i = 4 \\ 0 if & i ≥ 5 \end{matrix}

Example 4.10. Consider now the case A = F_{2}, the free group on two generators. Note that the pure braid group P_{3} ≅ ℤ × F_{2}. There is a fibration sequence

SU(2) × SU(2) − 𝕋^{2} → (SU(2))^{3} − Hom(ℤ × F_{2}, SU(2)) → 𝕊^{2}

which can be used to compute the cohomology of the complement.

Proposition 4.11. The Serre spectral sequence for the fibration above collapses at E_{2} and the cohomology of the total space is torsion–free with Poincaré series

(1 + t^{2})(1 + 3t^{3} + 2t^{4}) = 1 + t^{2} + 3t^{3} + 2t^{4} + 3t^{5} + 2t^{6}.

Using duality, it follows that

Theorem 4.12. H^{∗}(Hom(ℤ × F_{2}, SU(2)) is torsion free, with Poincaré series

1 + 2t^{2} + 6t^{3} + 2t^{4} + t^{5} + 2t^{6}.

The case for the complement of Hom(ℤ × F_{r}, SU(2)) can be handled similarly. Let p_{r}(t) denote the Poincaré series for the rational cohomology of SU(2)^{r} − 𝕋^{r}, then

Theorem 4.13. The spectral sequence for the fibration

SU(2)^{r} − 𝕋^{r} → SU(2)^{r+1} − Hom(ℤ × F_{r}, SU(2)) → 𝕊^{2}

collapses at E_{2} and the total space has torsion–free homology and Poincaré series equal to (1 + t^{2})p_{r}(t).

Note: p_{r}(t) is not hard to compute. Using duality it should be possible to obtain the Betti numbers for Hom(ℤ × F_{r}, SU(2)).

5. The Variety of Commuting Triples in SU(2)

This section will focus on the case of commuting triples in SU(2). As before it seems best to first consider its complement. Let X[m] denote the space of non–commuting m–tuples,

X[m] = (SU(2))^{m} − Hom(ℤ^{m}, SU(2)).

Notice that the symmetric group Σ_{m} acts naturally on both the space of commuting and non–commuting m–tuples. Moreover it has been shown that X[2] is homotopy equivalent to SO(3).

Let {i, j, k} = {1, 2, 3} and denote

X_{j,k} = {(z_{1}, z_{2}, z_{3}) ∈ (SU(2))^{3} | z_{j}z_{k} ≠ z_{k}z_{j}}.

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 20

Note that X[3] = X_{1,2} ∪ X_{1,3} ∪ X_{2,3}. The cohomology of X[3] will be computed by using this decomposition. Note that each of the three pieces is homeomorphic to X[2] × SU(2), and that Σ_{3} permutes these three pieces. The triple intersection consists of all pairwise non–commuting triples, and is invariant under the action of the symmetric group. Using results in the previous section, it follows that

X_{i,j} ≅ X[2] × SU(2) ≃ SO(3) × SU(2).

The cohomology of the intersections is computed next. Let X_{123} = X_{1,2} ∩ X_{1,3} ∩ X_{2,3}.

Proposition 5.1. The projection onto the j, k coordinates π_{jk} : X_{j,k} → X[2] restricts to fibrations

F_{i} → X_{j,k} ∩ X_{i,k} → X[2]
E_{i} → X_{123} → X[2]

where the fibers can be described as

F_{i} ≅ SU(2) − Z_{G}(z_{k}) ≅ SU(2) − 𝕊^{1} ≃ 𝕊^{1}
E_{i} ≅ SU(2) − [Z_{G}(z_{k}) ∪ Z_{G}(z_{j})] ≅ SU(2) − (𝕊^{1} ∪ 𝕊^{1}) ≃ 𝕊^{1} ⋁ 𝕊^{1} ⋁ 𝕊^{1}.

The map X[2] → X_{123} given by ψ(z_{j}, z_{k})_{j} = z_{j}, ψ(z_{j}, z_{k})_{k} = z_{k}, ψ(z_{j}, z_{k})_{i} = z_{j}z_{k} defines a section for all three bundles.

Proof. Given a triple z_{i}, z_{j}, z_{k} of pairwise non–commuting elements, its projection onto the j, k coordinates will be all non–commuting pairs. Further the fiber over any such pair (z_{j}, z_{k}) can be described as elements z_{i} ∈ SU(2) such that z_{i} is not in the centralizer of z_{j}, nor in the centralizer of z_{k}. Now given that z_{j}, z_{k} do not commute, they are both regular elements, and thus their centralizers are maximal tori (in this case isomorphic to a circle 𝕊^{1}) which intersect exactly at the center of SU(2), namely {±I}. Hence the fiber E_{i} is the complement in SU(2) of a union of two unknotted circles intersecting at two points. It is easy to see that this has the homotopy type of a wedge of three circles. Similarly the fiber F_{i} can be identified with the elements z_{i} ∈ SU(2) − Z_{G}(z_{k}) ≃ 𝕊^{1}.

Note that π_{j,k} restricted to X_{123} and X_{j,k} ∩ X_{i,k} respectively still maps onto X[2]. Furthermore the fibers are all homeomorphic and the bundles are locally trivial–hence they are fibrations.

Finally note that if z_{j} and z_{k} do not commute, then z_{j} cannot commute with z_{j}z_{k} and z_{k} cannot commute with z_{j}z_{k}; hence ψ(z_{j}, z_{k}) defines a non–commuting triple and hence a section for all three bundles. □

In what follows it will be shown that in fact these bundles are fibre homotopy equivalent to product bundles. Define a map

R : SO(3) × SU(2) → SU(2)^{3}

by the formula

R(ρ, z) = (i⃗^{ρ}, j⃗^{ρ}, z^{ρ})

where z^{ρ} = ρ · z · ρ^{−1} and i⃗, j⃗ denote the usual elements in the quaternions.

Proposition 5.2. The map

R : SO(3) × SU(2) → SU(2)^{3}

restricts to fibre homotopy equivalences

(1)

α : SO(3) × SU(2) → X_{1,2},

(2)

β : SO(3) × (SU(2) − [Z_{G}(i⃗)]) → X_{1,2} ∩ X_{1,3}

and (3)

γ : SO(3) × (SU(2) − [Z_{G}(i⃗) ∪ Z_{G}(j⃗)]) → X_{123}.

Proof. By Proposition 5.1, the projection onto the j, k coordinates π_{jk} : X_{j,k} → X[2] restricts to fibrations

F_{i} → X_{j,k} ∩ X_{i,k} → X[2]
E_{i} → X_{123} → X[2]

where the fibers are given by

(1)

F_{i} ≅ SU(2) − Z_{G}(z_{k}) ≅ SU(2) − 𝕊^{1}

which is homotopy equivalent to 𝕊^{1} and (2)

E_{i} ≅ SU(2) − [Z_{G}(z_{k}) ∪ Z_{G}(z_{j})] ≅ SU(2) − (𝕊^{1} ∪ 𝕊^{1})
which is homotopy equivalent to 𝕊^{1} ⋁ 𝕊^{1} ⋁ 𝕊^{1}.

Notice that the map R : SO(3) × SU(2) → SU(2)^{3} restricts to

α : SO(3) × SU(2) → X_{1,2}.

By Example 4.4, the composite λ : SO(3) → X[2] given by

SO(3) × {1} ⟶ SO(3) × SU(2) ⟶^{α} X_{1,2} ⟶^{project} X[2]

is a homotopy equivalence.

Since α(ρ, z) = (i⃗^{ρ}, j⃗^{ρ}, z^{ρ}), it follows that α induces a morphism of fibrations which, by inspection, gives a homotopy equivalence on the base λ : SO(3) → X[2]. In addition, consider the points (ρ, z) with z ∈ SU(2) in a fixed fibre over ρ ∈ SO(3). The induced map on fibres sends z to z^{ρ} which is a homeomorphism on each fibre SU(2). Thus the induced map of each fibre is an equivalence.

The map R restricts to

β : SO(3) × (SU(2) − [Z_{G}(i⃗)]) → X_{1,2} ∩ X_{1,3}

defined by β(ρ, z) = (i⃗^{ρ}, j⃗^{ρ}, z^{ρ}) which induces a morphism of fibrations

SU(2) − [Z_{G}(i⃗)]  ⟶^{β̄}  F_{p}
↓  ↓
SO(3) × (SU(2) − [Z_{G}(i⃗)]) ⟶^{β} X_{1,2} ∩ X_{1,3}
↓  ↓
SO(3)  ⟶^{λ} X[2]

where F_{p} denotes the fibre over a point p in the image of the projection X_{1,2} ∩ X_{1,3} → X[2] with induced maps on fibres denoted β̄ : SU(2) − [Z_{G}(i⃗)] → F_{p}.

This morphism of fibrations induces the homotopy equivalence on the base given by λ : SO(3) → X[2]. Furthermore, the induced map on fibres SU(2) − [Z_{G}(i⃗)] over the point ρ ∈ SO(3) sends a point z to z^{ρ} which is a homeomorphism

SU(2) − [Z_{G}(i⃗)] → SU(2) − [Z_{G}(i⃗^{ρ})]

as Z_{G}(z) is given by the points a + bz for a^{2} + b^{2} = 1.

Finally, the map R restricts to

γ : SO(3) × (SU(2) − [Z_{G}(i⃗) ∪ Z_{G}(j⃗)]) → X_{123}

defined by γ(ρ, z) = (i⃗^{ρ}, j⃗^{ρ}, z^{ρ}). This map induces a morphism of fibrations by inspection as well as giving the homotopy equivalence on the base λ : SO(3) → X[2] and a homeomorphism SU(2) − [Z_{G}(i⃗) ∪ Z_{G}(j⃗)] → SU(2) − [Z_{G}(i⃗^{ρ}) ∪ Z_{G}(j⃗^{ρ})].

The proposition follows.

□

Using analogous methods it is possible to establish the same result for all of the intersections:

Corollary 5.3. For {i, j, k} = {1, 2, 3} we have homotopy equivalences

X_{i,k} ∩ X_{j,k} ≃ SO(3) × 𝕊^{1}.

Now using the three inclusions ρ_{k} : X_{123} → X_{i,k} ∩ X_{j,k} it is possible to find a basis {e_{1}, e_{2}, e_{3}} for H^{1}(X_{123}, ℤ) such that if u ∈ H^{1}(X_{i,k} ∩ X_{j,k}, ℤ) is the canonical generator, then ρ_{k}^{∗}(u) = e_{k}. As a consequence of this analysis it follows that the alternating sum of the maps induced by inclusions in cohomology induces an isomorphism:

H^{1}(X_{1,2} ∩ X_{1,3}, ℤ) ⊕ H^{1}(X_{1,2} ∩ X_{2,3}, ℤ) ⊕ H^{1}(X_{1,3} ∩ X_{2,3}, ℤ) ≅ H^{1}(X_{123}, ℤ).

Consider the Mayer-Vietoris spectral sequence in integral cohomology associated to the covering X = X_{1,2} ∪ X_{1,3} ∪ X_{2,3}. It will have as its E_{1} term

E_{1}^{p,q} ≅ \{ \begin{matrix}H^{q}(X_{1,2}, ℤ) ⊕ H^{q}(X_{1,3}, ℤ) ⊕ H^{q}(X_{2,3}, ℤ) if & p = 0 \\ H^{q}(X_{1,2} ∩ X_{1,3}, ℤ) ⊕ H^{q}(X_{1,2} ∩ X_{2,3}, ℤ) ⊕ H^{q}(X_{1,3} ∩ X_{2,3}, ℤ) if & p = 1 \\ H^{q}(X_{123}, ℤ) if & p = 2 \\ 0 if & p ≥ 3 \end{matrix}

The differentials on the E_{1}–page are determined as alternating sums of the inclusion maps. From the previous observation d_{1}^{1,q} : E_{1}^{1,q} → E_{1}^{2,q} is an epimorphism for all q ≥ 0. Hence there are no higher differentials in the spectral sequence, which collapses at E_{2}, to yield

Theorem 5.4.

H^{∗}(X[3], ℤ) = H^{∗}((SU(2))^{3} − Hom(ℤ^{3}, SU(2)), ℤ) ≅ H^{∗}([⋁_{i=1}^{3} SU(2)] × SO(3), ℤ).

Once again using the fact that the spaces of homomorphisms under consideration are locally contractible, and applying Poincaré–Lefschetz duality to the pair

(SU(2))^{3}, Hom(ℤ^{3}, SU(2))

yields (see [5], page 292) that for all i ≥ 0 there is an isomorphism

H^{i}(Hom(ℤ^{3}, SU(2)), ℤ) ≅ H_{9−i}((SU(2))^{3}, X[3]), ℤ)

where as before X[3] is the space of non–commuting triples, whose (co)homology has just been computed. The basic ingredient needed is to understand the map induced in homology by the inclusion X[3] ⊂ SU(2) × SU(2) × SU(2). This is determined by the inclusions X_{j,k} = X[2] × SU(2) ⊂ (SU(2))^{3} and this in turn by the inclusion X[2] ⊂ SU(2) × SU(2). However it has already been shown that up to homotopy there is a factorization

X[2] ⊂ 𝕊^{2} × 𝕊^{2} → SU(2) × SU(2)

where 𝕊^{2} is identified with matrices of trace zero. Hence X[2] ⊂ SU(2) × SU(2) is trivial in reduced homology, and taking the product with SU(2) it follows that the map in reduced homology induced by X_{j,k} ⊂ (SU(2))^{3} is only non–trivial on the three dimensional class corresponding to the sphere, which is mapped via the identity. Putting these together, and observing that the sphere appearing in the X_{j,k}’s rotates around the three spheres in the product leads to

Lemma 5.5. The map in reduced integral homology induced by the inclusion X[3] ⊂ (SU(2))^{3} is a surjection in dimension equal to three, and zero everywhere else.

Applied to the long exact sequence in homology for the pair ((SU(2))^{3}, X[3]), this yields the following computation:

Theorem 5.6. The integral cohomology of the space of (ordered) commuting triples in SU(2) is given by

H^{i}(Hom(ℤ ⊕ ℤ ⊕ ℤ, SU(2)), ℤ) ≅ \{ \begin{matrix}ℤ if & i = 0 \\ 0 if & i = 1 \\ ℤ ⊕ ℤ ⊕ ℤ if & i = 2 \\ ℤ ⊕ ℤ ⊕ ℤ if & i = 3 \\ ℤ/2ℤ ⊕ ℤ/2ℤ ⊕ ℤ/2ℤ if & i = 4 \\ ℤ if & i = 5 \\ 0 if & i = 6 \\ ℤ/2ℤ if & i = 7 \\ 0 if & i ≥ 8 \end{matrix}

See [3] for a different perspective on commuting elements in compact Lie groups.

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 25

6. Degenerate Subspaces and Stable Structure for Commuting Elements

Natural subspaces of Hom(ℤ^{n}, G) arise from the fat wedge filtration of the product G^{n} where the base-point of G is 1_{G} with F_{j}G^{n} given by the subspace of G^{n} with at least j coordinates equal to 1_{G}. Define subspaces of Hom(ℤ^{n}, G) by the formula

S_{n}(j, G) = Hom(ℤ^{n}, G) ∩ F_{j}G^{n}

and let S_{n}(G) = S_{n}(1, G). There is an induced filtration

S_{n}(n, G) ⊂ S_{n}(n − 1, G) ⊂ ⋯ ⊂ S_{n}(1, G) ⊂ Hom(ℤ^{n}, G).

The key property required is the following.

Definition 6.1. A Lie group G is said to have cofibrantly commuting elements if the natural inclusions

I_{j} : S_{n}(j, G) → S_{n}(j − 1, G)

are cofibrations with cofibre S_{n}(j − 1, G)/S_{n}(j, G) for all n and j for which both spaces are non-empty.

A result due to Steenrod [16] is that if the pair (X, A) is an NDR-pair, then the map A → X is a cofibration with cofibre X/A. In section 7, it will be verified that if G is a closed subgroup of GL(m, ℂ), then the pairs (S_{n}(j − 1, G), S_{n}(j, G)) are all NDR pairs and hence that these groups G have cofibrantly commuting elements. Given this technical condition, it will follow that the spaces Hom(ℤ^{n}, G) naturally split after a single suspension, where the summands which appear are the suspensions of the successive quotients S_{n}(j−1, G)/S_{n}(j, G) arising from the filtration. These in turn can be expressed using quotients of the form Hom(ℤ^{q}, G)/S_{q}(G).

The decompositions of Σ(Hom(ℤ^{n}, G)) will make use of the following lemma.^{3}

Lemma 6.2. Let

A ⟶^{i} B ⟶^{p} C

be a cofibration of path-connected spaces of the homotopy type of a CW-complex that is split after one suspension, i.e. assume given a map

r : Σ(B) → Σ(A)

such that r ∘ Σ(i) is homotopic to 1_{Σ(A)}. Then the map

Σ(B) \underrightarrow{r ⋁ Σ(p)} Σ(A) ⋁ Σ(C)

3Recall that if X and Y are pointed topological spaces, then X ⋁ Y denotes their one point union in the Cartesian product X × Y, while ΣX denotes the suspension of X.

is a homotopy equivalence.

Proof. Consider the morphism of cofibrations given by

Σ(A) ⟶^{Σ(i)}  Σ(B)  ⟶^{Σ(p)} Σ(C)
↓^{1}  ↓^{r ⋁ Σ(p)}  ↓^{1}
Σ(A) ⟶^{Σ(i)} Σ(A) ⋁ Σ(C) ⟶^{project} Σ(C)

Thus r ⋁ Σ(p) is a homology isomorphism. This suffices.

□

Assume that

i : A → B

is a cofibration. The crux of the matter is to give a choice of map r : Σ(B) → Σ(A) as given in the lemma. An important example which will be made use of is the inclusion X ⋁ Y → X × Y, where X and Y are pointed CW –complexes. In this case the splitting map

π : Σ(X × Y) → Σ(X) ⋁ Σ(Y)

is given by the following composite

Σ(X × Y) ⟶^{pinch} Σ(X × Y) ⋁ Σ(X × Y) \underrightarrow{Σ(p_{X}) ⋁ Σ(p_{Y} )} Σ(X) ⋁ Σ(Y)

where p_{X} : X × Y → X is the natural projection map.

Consider the inclusion ⋁^{n} G → Hom(ℤ, G) and its mapping cone, denoted A_{n}(G). By construction this will be a cofibration sequence. The composite

⋁^{n} G ⟶ Hom(ℤ^{n}, G) ⟶ G^{n},

factors the inclusion of the wedge ⋁^{n} G in the product G^{n}. This map is split after a single suspension; hence it follows that after one suspension, the natural cofibration ⋁^{n} G → Hom(ℤ^{n}, G) → A_{n}(G) is also split. From this it follows that if G is any Lie group, then there is a homotopy equivalence

Σ(Hom(ℤ^{n}, G)) ≃ Σ(⋁^{n} G) ⋁ Σ(A_{n}(G)).

However, the geometric identification of A_{n}(G) has not been made precise, even for n = 2. This is precisely what is desired for certain important examples, assuming the hypothesis that G has cofibrantly commuting elements.

In what follows consideration will be given to the successive quotients K(n, q, G) = S_{n}(n− q, G)/S_{n}(n − q + 1), and it will be important to have a precise description of them.

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 27

Theorem 6.3. The quotient

K(n, q, G) = S_{n}(n − q, G)/S_{n}(n − q + 1, G)

is homeomorphic to the subspace of the product

\begin{matrix}( & {n \atop q} & ) \\ ∏ \end{matrix} Hom(ℤ^{q}, G)/S_{q}(G)

given by the wedge

\begin{matrix}( & {n \atop q} & ) \\ ⋁ \end{matrix} Hom(ℤ^{q}, G)/S_{q}(G).

Thus the suspension ΣK(n, q, G) is a retract of Σ(∏^{\binom{n}{q}} Hom(ℤ^{q}, G)/S_{q}(G)).

Proof. Given an ordered k element subset I = (n_{1}, n_{2}, ⋯, n_{k}) of {1, 2, ⋯, n} with

1 ≤ n_{1} < n_{2} < ⋯ < n_{k} ≤ n,

there are associated projection maps

P_{I} : Hom(ℤ^{n}, G) → Hom(ℤ^{k}, G).

There are embeddings

S_{I} : Hom(ℤ^{k}, G) → Hom(ℤ^{n}, G)

by inserting the identity 1_{G} in the coordinates not equal to n_{1}, n_{2}, ⋯, n_{k}. The composite

Hom(ℤ^{k}, G) ⟶^{S_{I}} Hom(ℤ^{n}, G) ⟶^{P_{I}} Hom(ℤ^{k}, G)

is the identity. Thus if k < n, Hom(ℤ^{k}, G) is a retract of Hom(ℤ^{n}, G). Notice that an analogous statement is satisfied for any simplicial space.

These projection maps can be assembled to yield a map

Hom(ℤ^{n}, G) → \begin{matrix}( & {n \atop q} & ) \\ ∏ \end{matrix} Hom(ℤ^{q}, G)/S_{q}(G).

Note that the subspace S_{n}(n − q + 1, G) will map to a point, and that S_{n}(n − q, G) maps onto the wedge

\begin{matrix}( & {n \atop q} & ) \\ ⋁ \end{matrix} Hom(ℤ^{q}, G)/S_{q}(G)

as a subspace of the product. It is easy to see that this induces a continuous, closed bijection

K(n, q, G) → \begin{matrix}( & {n \atop q} & ) \\ ⋁ \end{matrix} Hom(ℤ^{q}, G)/S_{q}(G)

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 28

and the result follows.

□

In order to proceed any further, the aforementioned technical condition on the target group G must be assumed, i.e. that G has cofibrantly commuting elements.

Theorem 6.4. Let G be a Lie group which has cofibrantly commuting elements, then the inclusions

S_{n}(q, G) → S_{n}(q − 1, G)

split after a single suspension.

Proof. It suffices to show that

S_{n}(n − q, G) → Hom(ℤ^{n}, G)

splits after a single suspension. Induction on q will be used. For q = 1, S_{n}(n − 1, G) = ⋁^{n} G and the splitting exists by using the inclusion into the product ∏^{n} G. Now assume it for q − 1 and prove it for q.

There is a commutative diagram

S_{n}(n − q + 1, G) ⟶^{1}  S_{n}(n − q + 1, G)
↓  ↓
S_{n}(n − q, G) ⟶  Hom(ℤ^{n}, G)
↓  ↓
K(n, q, G) ⟶^{γ} Hom(ℤ^{n}, G)/S_{n}(n − q + 1, G)

By the inductive hypotheses, the vertical columns split after suspending once. Here the fact that the two columns are cofibration sequences, arising from NDR pairs is being used. It is only necessary to show that the bottom map splits after a single suspension. As before, a map can be constructed

Hom(ℤ^{n}, G) → \begin{matrix}( & {n \atop q} & ) \\ ∏ \end{matrix} Hom(ℤ^{q}, G)/S_{q}(G)

using the projections. Now note that once again S_{n}(n − q + 1, G) maps to a single point, hence it is well defined on the quotient. The composition

ΣHom(ℤ^{n}, G)/S_{n}(n − q + 1, G) → Σ \begin{matrix}( & {n \atop q} & ) \\ ∏ \end{matrix} Hom(ℤ^{q}, G)/S_{q}(G) → ΣK(n, q, G)

can be used to split our map, hence completing the proof. □

Theorem 6.5. If G is a Lie group which has cofibrantly commuting elements, then there are homotopy equivalences

Σ(Hom(ℤ^{n}, G)) ≃ ⋁_{1≤k≤n} Σ(\begin{matrix}( & {n \atop k} & ) \\ ⋁ \end{matrix} Hom(ℤ^{k}, G)/S_{k}(G)).

Proof. There is a filtration of Hom(ℤ^{n}, G):

S_{n}(n, G) ⊂ S_{n}(n − 1, G) ⊂ ⋯ ⊂ S_{n}(1, G) ⊂ Hom(ℤ^{n}, G)

where each successive inclusion is a cofibration which is split after a single suspension. By induction there is a splitting

ΣHom(ℤ^{n}, G) ≃ ⋁_{1≤q≤n} ΣK(n, q, G)

whence the result follows. □

Remark. The proof given here in the special setting of Σ(Hom(ℤ^{n}, G)) applies to the following general setting of a simplicial space satisfying a certain cofibration condition. Let X_{∗} denote any simplicial space with n-th space given by X_{n}. Define

s_{q}(X_{n}) = ∪_{j_{1}>j_{2}>···>j_{q}≥0}s_{j_{1}}s_{j_{2}} ⋯ s_{j_{q}}(X_{n−q}).

Theorem 6.6. If X_{∗} is a simplicial space such that the maps

s_{q}(X_{n}) → s_{q−1}(X_{n})

are cofibrations for all q, then there are homotopy equivalences

Σ(X_{n}) ≃ ⋁_{0≤k≤n} Σ(\begin{matrix}( & {n \atop k} & ) \\ ⋁ \end{matrix} X_{k}/s_{k}(X_{k})).

For the sake of concreteness, a proof in the special setting of Theorem 6.5 is completed below, while the general case will appear elsewhere. This result, relatively direct, is useful in the sense that the cohomology groups of the space Hom(ℤ^{n}, G) are isomorphic to those of the space ⋁_{1≤k≤n} Σ(⋁^{\binom{n}{k}} Hom(ℤ^{k}, G)/S_{k}(G)) for any cohomology theory. For the purposes here, the main content is checking that a Lie group G has cofibrantly commuting elements.

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 30

Examples concerning these splittings for Hom(ℤ^{n}, SU(2)) are given next: assume throughout that SU(2) has cofibrantly commuting elements. The next examples follow from the stable decompositions above in conjunction with the Mayer-Vietoris spectral sequence computations given in the previous section.

Example 6.7. Hom(ℤ^{2}, SU(2)):

The assumption is being made that for G = SU(2), the inclusion

G ⋁ G → Hom(ℤ^{2}, G)

is a cofibration sequence with cofiber the natural quotient; then there is a homotopy equivalence A_{2}(G) ≃ Hom(ℤ^{2}, G)/G ⋁ G.

Since the map G ⋁ G → Hom(ℤ^{2}, G) is an embedding, it follows that

G × G − Hom(ℤ^{2}, G) ⊂ G × G − G ⋁ G

is an embedding. Recall that there is an embedding SO(3) → G×G−Hom(ℤ^{2}, G) which is a homotopy equivalence, and that the image of SO(3) has a neighborhood which is a product bundle SO(3) × ℝ^{3}. It follows that A_{2}(SU(2)) is homotopy equivalent to the Spanier-Whitehead dual 𝕊^{6} − SO(3).

Proposition 6.8. If G = SU(2), there is a homotopy equivalence

Σ(Hom(ℤ^{2}, G)) ≃ Σ(G ⋁ G) ⋁ Σ(𝕊^{6} − SO(3)).

From this a cohomology calculation follows:

H^{i}(Hom(ℤ^{2}, SU(2)), ℤ) = H^{i}(⋁^{2} SU(2) ⋁(𝕊^{6} − SO(3)), ℤ).

Note that this agrees with the previous computation.

Example 6.9. Hom(ℤ^{3}, SU(2)):

After a single suspension, Hom(ℤ^{3}, SU(2)) is homotopy equivalent to

⋁^{3} SU(2) ⋁ S_{3}(1, SU(2))/S_{3}(2, SU(2)) ⋁ Hom(ℤ^{3}, SU(2))/S_{3}(1, SU(2))

Now

S_{3}(1, SU(2))/S_{3}(2, SU(2)) = ⋁^{3} Hom(ℤ^{2}, SU(2))/S_{2}(1, SU(2)) ≃ ⋁^{3}[𝕊^{6} − SO(3)].

By our homological computations, Hom(ℤ^{3}, SU(2))/S_{3}(SU(2)) is homotopy equivalent to SU(2) ∧ (S^{6} − SO(3)). Hence it follows that

ΣHom(ℤ^{3}, SU(2)) ≃ Σ ⋁^{3} SU(2) ⋁[Σ ⋁^{3}(𝕊^{6} − SO(3))] ⋁[ΣSU(2) ∧ (𝕊^{6} − SO(3)].

Example 6.10. : Hom(ℤ^{n}, SU(2)):

The first three terms in the decomposition can be determined, and so ΣHom(ℤ^{n}, SU(2)) splits off a summand of the form

Σ ⋁^{n} SU(2) ⋁[Σ \begin{matrix}( & {n \atop 2} & ) \\ ⋁ \end{matrix}(𝕊^{6} − SO(3))] ⋁[\begin{matrix}( & {n \atop 3} & ) \\ ⋁ \end{matrix} ΣSU(2) ∧ (𝕊^{6} − SO(3))],

The remaining stable summands are not yet understood.

7. NDR Pairs and Cofibrations

Useful techniques in the proofs of the stable decompositions above arise from information concerning the geometry of certain Lie groups, centralizers of elements, and their relationship to the homotopy theoretic notion of an NDR-pair. Recall Steenrod’s definition [16] that the pair (X, A) is said to be an NDR-pair provided there exist continuous maps

u : X → [0, 1],

and

h : [0, 1] × X → X

which satisfy the following properties.

(1) A = u^{−1}(0),

(2) h(0, x) = x for all x in X,

(3) h(t, a) = a for all a in A, and

(4) h(1, v) is in A for all v in u^{−1}([0, 1)). Furthermore, if (X, A) is an NDR-pair, then the natural inclusion A → X is a cofibration with cofibre given by the quotient space X/A.

Steenrod proves the following Lemma [16] (stated as Lemma 6.3), and with the extension in the second part due to May [14] (page 164).

Lemma 7.1. (1) Let (h, u) and (j, v) represent (X, A), and (Y, B) as NDR-pairs. Then (k, w) represents the product pair

(X, A) × (Y, B) = (X × Y, X × B ∪ A × Y) as an NDR-pair, where w(x, y) = min(u(x), v(y)) and

k(t, x, y) = \begin{cases}(h(t, x), j([u(x)/v(y)]t, y)) & if v(y) ≥ u(x), \\ (h([v(y)/u(x)]t, x), j(t, y)) & if u(x) ≥ v(y). \end{cases}

(2) Let (h, u) represent (X, A) as an NDR-pair. Then (h_{j}, u_{j}) represents

(X, A)^{j} = (X^{j}, ∪_{1≤i≤j}X^{i−1} × A × X^{j−i})

as an NDR-pair (which is Σ_{j}-equivariant) where

u_{j}(x_{1}, x_{2}, ⋯, x_{j}) = min(u(x_{1}), u(x_{2}), ⋯, u(x_{j})),

and

h_{j}(t, x_{1}, x_{2}, ⋯, x_{j}) = (h(t_{1}, x_{1}), ⋯, h(t_{j}, x_{j}))

with

t_{i} = \begin{cases}t · min_{j≠i}(u(x_{j})/u(x_{i})) & if some u(x_{j}) < u(x_{i}), j ≠ i, \\ t & if all u(x_{j}) ≥ u(x_{i}), j ≠ i. \end{cases}

This lemma can be used to check that (Hom(ℤ^{n}, G), S_{n}(G)) is an NDR-pair. However, a modification is required to prove the full stable splitting in Theorem 1.6. That result requires that all the (S_{n}(q, G), S_{n}(q+1, G)) are NDR-pairs as long as S_{n}(q+1, G) is non-empty. Thus, let (X, A) denote an NDR-pair with F_{r}X^{n} the subspace of X^{n} given by {(x_{1}, x_{2}, ⋯, x_{n})} for which at least r of the x_{j} are in A.

Lemma 7.2. Let (h, u) represent (X, A) as an NDR-pair. Fix integers 1 ≤ r ≤ n. Then (h_{n}, u_{r,n}) represents (X^{n}, F_{r}X^{n}) as a Σ_{n}-equivariant NDR-pair where

u_{r,n}(x_{1}, x_{2}, ⋯, x_{n}) = 1/r · min_{1≤i_{1}<i_{2}<···<i_{r}≤n}(u(x_{i_{1}}) + u(x_{i_{2}}) + ⋯ + u(x_{i_{r}})),

and

h_{n}(t, x_{1}, x_{2}, ⋯, x_{j}) = (h(t_{1}, x_{1}), ⋯, h(t_{n}, x_{n}))

with

t_{i} = \begin{cases}t · min_{i≠n}(u(x_{m})/u(x_{i})) & if some u(x_{m}) < u(x_{i}), m ≠ i, \\ t & if all u(x_{m}) ≥ u(x_{i}), m ≠ i. \end{cases}

Proof. Notice that (1) u_{r,n} is continuous,

(2) u_{r,n}(x_{1}, x_{2}, ⋯, x_{n}) = 0 if and only if u(x_{i_{1}}) + u(x_{i_{2}}) + ⋯ + u(x_{i_{r}}) = 0 for some sequence 1 ≤ i_{1} < i_{2} < ⋯ < i_{r} ≤ n. Since 0 ≤ u(x) ≤ 1, it follows that u(x_{i_{t}}) = 0 for 1 ≤ t ≤ r, and so x_{i_{t}} is in A. Hence u_{r,n}^{−1}(0) = F_{r}X^{n}.

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 33

Finally, notice that

(1) h_{n}(−, −) is continuous,

(2) h_{n}(0, x_{1}, x_{2}, ⋯, x_{n}) = (x_{1}, x_{2}, ⋯, x_{n}) as h(0, x) = x,

(3) h_{n}(t, x_{1}, x_{2}, ⋯, x_{n}) = (x_{1}, x_{2}, ⋯, x_{n}) if (x_{1}, x_{2}, ⋯, x_{n}) is in F_{r}X^{n} for all t in [0, 1] as h(t, x) is in A for x in A, and

(4) h_{n}(1, x_{1}, x_{2}, ⋯, x_{n}) is in F_{r}X^{n} if (x_{1}, x_{2}, ⋯, x_{n}) is in u_{r,n}^{−1}([0, 1)).

The lemma follows. □

The next result provides a practical criterion used in the next section for testing whether certain Lie groups have cofibrantly commuting elements.

Theorem 7.3. Let (h, u) be a representation of (G, {1_{G}}) as an NDR-pair with the additional property that for each g ≠ 1_{G} in G with 0 ≤ t < 1, the centralizer of h(t, g), Z(h(t, g)), equals the centralizer of g, Z(g). If S_{n}(r, G) is non-empty, then (h_{n}, u_{r,n}) the representation of (G^{n}, F_{r}G^{n}) as a Σ_{n}-equivariant NDR-pair as given in Lemma 7.2, restricts to give a representation of (S_{n}(r−1, G), S_{n}(r, G)) as a Σ_{n}-equivariant NDR pair. Thus G has cofibrantly commuting elements.

Proof. The hypothesis of Theorem 7.3 is that (h, u) is a representation of (G, {1_{G}}) as an NDR-pair with the additional property that for each g ≠ 1_{G} in G with 0 ≤ t < 1, the centralizer of h(t, g) equals the centralizer of g.

By Lemma 7.2, the pair of maps (h_{n}, u_{r,n}) represents (G^{n}, F_{r}G^{n}) as an NDR-pair where

u_{r,n}(x_{1}, x_{2}, ⋯, x_{n}) = 1/r · min_{1≤i_{1}<i_{2}<···<i_{r}≤n}(u(x_{i_{1}}) + u(x_{i_{2}}) + ⋯ + u(x_{i_{r}})),

and h_{n}(t, x_{1}, x_{2}, ⋯, x_{j}) = (h(t_{1}, x_{1}), ⋯, h(t_{n}, x_{n})) with

t_{i} = \begin{cases}t · min_{i≠m}(u(x_{m})/u(x_{i})) & if some u(x_{m}) < u(x_{i}), m ≠ i, \\ t & if all u(x_{m}) ≥ u(x_{i}), m ≠ i. \end{cases}

Thus the homotopy defined by h_{n}(t, x_{1}, x_{2}, ⋯, x_{n}) = (h(t_{1}, x_{1}), ⋯, h(t_{n}, x_{n})) satisfies the following properties:

(1) The centralizer of h(t, g) equals the centralizer of g for all g ≠ 1_{G} and 0 ≤ t < 1.

(2) The element h_{n}(t, x_{1}, x_{2}, ⋯, x_{n}) is in Hom(ℤ^{n}, G).

(3) The homotopy h_{n}(t, x_{1}, x_{2}, ⋯, x_{n}) preserves the subspaces S_{n}(r, G) and S_{n}(r−1, G). Thus the function h_{n} restricts to

h_{n} : [0, 1] × S_{n}(r, G) → S_{n}(r, G)

COMMUTING ELEMENTS AND SPACES OF HOMOMORPHISMS 34

and there is a commutative diagram

[0, 1] × S_{n}(r, G) ⟶^{h_{n}} S_{n}(r, G)
↓^{1×I_{r}}  ↓^{I_{r}}
[0, 1] × S_{n}(r − 1, G) ⟶^{h_{n}} S_{n}(r − 1, G)

for which I_{r} : S_{n}(r, G) → S_{n}(r − 1, G) denotes the natural inclusion. Since

u_{r,n}^{−1}(0) ∩ S_{n}(r − 1, G) = S_{n}(r, G),

the pair (h_{n}, u_{r,n}) restricts to a representation of (S_{n}(r − 1, G), S_{n}(r, G)) as an NDR pair. Thus G has cofibrantly commuting elements. □

8. Closed Subgroups of GL(n, ℂ) Have Cofibrantly Commuting Elements

In this section it will be shown that any closed subgroup of GL(n, ℂ) has cofibrantly commuting elements, as mentioned in the introduction (Theorem 1.5). Assume that G is any such group, with Lie algebra 𝒢. The adjoint representation

Ad : G → Aut(𝒢)

is defined by the equation Ad_{b}(y) = b · y · b^{−1} for b in G and y in 𝒢. Further assume that B_{ϵ}(0) is a ball of radius ϵ > 0 with center at the origin 0 in 𝒢 and closure B̄_{ϵ}(0) such that the restriction of the exponential map

exp : B̄_{ϵ}(0) → G

is a homeomorphism onto its’ image.

Remark. The map exp is a local homeomorphism in case G is a finite dimensional Lie group [8], but may not be an embedding on all of 𝒢 in case G is compact.

Lemma 8.1. Let y denote an element in B_{ϵ}(0) which is not 0. Then the centralizer of exp(y) in G is contained in the centralizer of exp(ty) in G for any real number t. Furthermore, if y ≠ 0 and 0 < t ≤ 1, the centralizers of exp(ty) and exp(y) in G are equal.

Proof. The following equation is classical [8]:

b · (exp(y)) · b^{−1} = exp(Ad_{b}(y))

where Ad_{b}(y) = b · (y) · b^{−1}.

Assume that exp(y) and b commute. Thus

exp(y) = b · (exp(y)) · b^{−1} = exp(Ad_{b}(y)).

Hence Ad_{b}(y) = b · y · b^{−1} = y as exp restricted to B_{ϵ}(0) is a homeomorphism. Thus b·(ty)·b^{−1} = ty for any scalar t as t is central. Furthermore, b·(exp(ty))·b^{−1} = exp(Ad_{b}(ty)) = exp(ty) and exp(ty) commutes with b. Thus the centralizer of exp(y) is contained in the centralizer of exp(ty) for all t.

Next assume that 0 < t ≤ 1 and consider the centralizer of exp(ty). Since ty is in B_{ϵ}(0) with ty ≠ 0, the analogous argument gives that the centralizer of exp(ty) is in the centralizer of exp(sty) for all real numbers s. The lemma follows by setting s = 1/t.

□

Define functions

u, s : G → [0, 1]

by the formulas

u(g) = \begin{cases}2|y|/ϵ & if g = exp(y) for g in exp(B̄_{ϵ/2}(0)), \\ 1 & if g is in G − exp(B_{ϵ/2}(0)) \end{cases}

and

s(g) = \begin{cases}1 & if g = exp(y) for g in exp(B̄_{ϵ/2}(0)), \\ 2 − u(g) & if g = exp(y) for g in exp(B̄_{ϵ}(0)) − exp(B_{ϵ/2}(0)), \\ 0 & if g is in G − exp(B_{ϵ}(0)). \end{cases}

Notice that the functions s, and u are both continuous functions. In addition, u^{−1}(0) = {1_{G}} and u^{−1}([0, 1)) = exp(B_{ϵ/2}(0)).

Next, define a function

h : [0, 1] × G → G

by the formula

h(t, g) = \begin{cases}exp((1 − t)y) & if g = exp(y) for g in exp(B̄_{ϵ/2}(0)), \\ exp((1 − s(g)t) · y) & if g = exp(y) for g in exp(B̄_{ϵ}(0)) − exp(B_{ϵ/2}(0)), and \\ g & if g is in G − B_{ϵ}(0). \end{cases}

Proposition 8.2. If G is any closed subgroup of GL(n, ℂ), then the pair (h, u) is a representation of (G, {1_{G}}) as an NDR pair. Furthermore, the representation (h, u) satisfies the additional condition that for each g ≠ 1_{G} in G with 0 ≤ t < 1, the centralizer of h(t, g), Z(h(t, g)), equals the centralizer of g, Z(g).

Proof. Observe that the homotopy h is continuous with u : G → [0, 1]. Furthermore (1) h(0, g) = g for all g in G,

(2) h(t, 1_{G}) = 1_{G} for all t in [0, 1] and

(3) u^{−1}([0, 1)) = exp(B_{ϵ/2}(0)), thus h(1, g) = 1_{G} if u(g) < 1

by a direct verification. Thus (G, {1_{G}}) is an NDR pair.

Notice that an inspection of the definition gives that the value of h(t, g) is either

(1) g,

(2) exp((1 − s)y) for y ≠ 0 and 1 − s ≠ 0 or

(3) 1_{G}.

It suffices to consider the centralizer of h(t, g) in the first two cases. Observe that in these cases, the centralizer of h(t, g) is equal to that of g by Lemma 8.1. Thus the representation (h, u) satisfies the additional condition that for each g ≠ 1_{G} in G with 0 ≤ t < 1, the centralizer of h(t, g), Z(h(t, g)), equals the centralizer of g, Z(g). The proposition follows. □

Theorem 8.3. If G is a closed subgroup of GL(n, ℂ), then G has cofibrantly commuting elements.

Proof. The previous proposition gives a verification of the hypotheses of Theorem 7.3 thus implying that if G is a closed subgroup of GL(n, ℂ), then G has cofibrantly commuting elements. The theorem follows. □

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Department of Mathematics University of British Columbia, Vancouver, B. C., Canada

 E-mail address: adem@math.ubc.ca

Department of Mathematics, University of Rochester, Rochester, NY 14627

 E-mail address: cohf@math.rochester.edu