ON ORBIT CONFIGURATION SPACES ASSOCIATED TO THE GAUSSIAN INTEGERS: HOMOTOPY AND HOMOLOGY GROUPS

F. R. COHEN* AND M. XICOTÉNCATL**

Abstract. The purpose of this article is to analyze several Lie algebras associated to “orbit configuration spaces” associated to the standard integral lattice ℤ + iℤ in the complex numbers. The Lie algebra obtained from the descending central series for the associated fundamental group is shown to be isomorphic, up to a regrading, to the Lie algebra obtained from the higher homotopy groups of “higher dimensional arrangements” modulo torsion. The resulting Lie algebras are similar to those studied by T. Kohno associated to elliptic KZ systems [7]. A question about the generality of this behavior is posed.

1. Introduction

The purpose of this article is to compare properties of certain Lie algebras obtained from (i) the descending central series for the fundamental group of certain choices of K(π, 1) spaces together with (ii) Lie algebras obtained from the classical higher homotopy groups of related loop spaces.

The evidence here and elsewhere suggests that in the special case of certain hyperplane arrangements, the Lie algebras encountered for the groups π, and those for the higher homotopy groups of “higher dimensional analogues” of hyperplane arrangements modulo torsion are isomorphic up to a regrading of the underlying Lie algebras. One “mantra” is that Vasilliev invariants for pure braids are determined by the higher homotopy groups of certain configuration spaces modulo torsion. This “mantra” is satisfied for certain other examples that are described next.

A more precise version of this conjecture is stated below for which several examples are already known. Namely, the homotopy groups of the loop space of the configuration space of

Date: November 1999.

1991 Mathematics Subject Classification. Primary: 55R80, 20F14; Secondary: 55N25, 55Q52.

*The first author was partially supported by the NSF.

**Supported by the CINVESTAV of the IPN and by the MPI für Mathematik.

2 F. R. COHEN* AND M. XICOTÉNCATL**

k points in an even dimensional euclidean space ℝ^{2m+2}, modulo torsion, admits the structure of a graded Lie algebra induced by the classical Samelson product. Apart from regrading, that Lie algebra is isomorphic to E_{0}^{∗}(P_{k}), the Lie algebra obtained from the descending central series for the pure k-stranded braid group that is ”universal” for the Yang-Baxter-Lie relations [3, 5]. Appealing to work of T. Kohno [7, 8, 9], these homotopy groups yield Vassiliev invariants for braids. By work of the second author [10], and D. Cohen [1], there are analogous results for certain choices of “orbit configuration spaces” of points in ℂ^{n} − {0} that are discussed in this article.

The main results here arise from choices of orbit configuration spaces as introduced by the second author in [10]. Given a manifold M on which the group G acts properly discontinuously, let F_{G}(M, k) be the orbit configuration space

F_{G}(M, k) = {(m_{1}, …, m_{k}) ∈ M^{k} | Gm_{i} ∩ Gm_{j} = ∅ if i ≠ j}.

Two examples of spaces M and groups G arise by considering (i) a parametrized lattice acting on the complex numbers ℂ so that the orbit space is an elliptic curve, and (ii) a discrete group acting properly discontinuously on the upper 1/2-plane H so that the orbit space is a complex curve. The orbit configuration spaces associated to the action of the standard integral lattice on the complex numbers is considered in this article.

Given a family of linear equations over the integers Σ_{1≤i≤k}a_{i}x_{i}, say Θ(k), let X(k, ℝ^{n}) denote the complement of the variety determined by Θ(k), V (Θ(k)), given by (ℝ^{n})^{k} − V (Θ(k)). Furthermore, assume that there are fibrations X(k, ℝ^{n}) → X(k − 1, ℝ^{n}) having sections with fibre given by ℝ^{n} − S_{k} where S_{k} is a discrete set. Consider the graded Lie algebra E_{0}^{∗}(π_{1}(X(k, ℝ^{2})))_{q}. Comparing answers for several known examples, suggests that E_{0}^{∗}(π_{1}(X(k, ℝ^{2}))) is isomorphic to the Lie algebra of primitive elements in the homology of ΩX(k, ℝ^{2q+2}) for q > 0. Furthermore, it is an easy exercise that the module of primitives is isomorphic to the Lie algebra E_{0}^{∗}(π_{∗}(X(k, ℝ^{2q+2}))) modulo torsion for q > 0.

There is an associated natural question that is stated roughly in the following paragraphs. Start with functors from vector spaces over ℝ with morphisms restricted to isometric embeddings, X(ℝ^{n}, k) where k is a natural number, with X(ℝ^{n}, 1) = ℝ^{n}, together with natural transformations X(ℝ^{n}, k) → X(ℝ^{n}, k − 1) such that these maps are fibrations with fibre ℝ^{n} − S_{k} where S_{k} is a discrete subspace of ℝ^{n} of fixed cardinality depending on k and where each fibration X(ℝ^{n}, k) → X(ℝ^{n}, k − 1) admits a section.

Next consider the Lie algebra obtained from the descending central series for the group G. For each strictly positive integer q, there is a canonical (and trivially defined) graded Lie algebra E_{0}^{∗}(G)_{q} attached to the one obtained from the descending central series for G, and which is defined as follows.

1. Fix a strictly positive integer q.

2. Let Γ^{n}(G) denote the n-th stage of the descending central series for G.

3. E_{0}^{2nq}(G)_{q} = Γ^{n}(G)/Γ^{n+1}(G),

4. E_{0}^{i}(G)_{q} = {0}, if i ≢ 0 mod 2q, and

5. the Lie bracket is induced by that for the associated graded for the Γ^{n}(G).

It seems likely that for many choices of X(ℝ^{2q+2}, k) the Lie algebra associated to G = π_{1}(X(ℝ^{2}, k)), satisifies the condition that it also gives the Lie algebra of primitive elements in the homology of certain loop spaces: E_{0}^{∗}(G)_{q} for q > 0 is isomorphic to the Lie algebra of primitive elements in the homology for the loop space ΩX(ℝ^{2q+2}, k) with q > 0. This last Lie algebra for the module of primitives is a graded free abelian group that is isomorphic to the Lie algebra given by the homotopy groups of ΩX(ℝ^{2q+2}, k) modulo torsion.

Theorem 1. Assume that n ≥ 3. Let X(ℝ^{n}, k) → X(ℝ^{n}, k−1) be a fibration which satisfies the following properties:

1. The fibre of X(ℝ^{n}, k) → X(ℝ^{n}, k − 1) is ℝ^{n} − S_{k} where S_{k} is a discrete subspace of ℝ^{n} of fixed cardinality depending on k,

2. Each fibration X(ℝ^{n}, k) → X(ℝ^{n}, k − 1) admits a cross-section, and

3. X(ℝ^{n}, 1) = ℝ^{n} with n ≥ 3.

Then

1. There is a homotopy equivalence ΩX(ℝ^{n}, k) → ∏_{1≤i≤k−1} Ω(ℝ^{n} − S_{i}).

2. The homology of ΩX(ℝ^{n}, k) is torsion free, and is isomorphic to ⨂_{1≤i≤k−1} H_{∗}(Ω(ℝ^{n} − S_{i})) as a coalgebra.

3. The module of primitives in the integer homology of ΩX(ℝ^{n}, k) is isomorphic to E_{0}^{∗}(π_{∗}(ΩX(ℝ^{n}, k))) modulo torsion as a Lie algebra.

4 F. R. COHEN* AND M. XICOTÉNCATL**

The proof of this theorem is an exercise that is given at the end of this article. The main purpose of this article is to study one example related to the theorem above given by the space F_{G}(M, k) in the case when M = ℂ, the complex numbers, and G is the integral lattice L = ℤ+iℤ, acting by translation on ℂ. One of the consequences of the theorem below is that the Lie algebra obtained from the fundamental group of the associated orbit configuration space also gives the Lie algebra obtained form the higher homotopy groups of the “higher dimensional analogues” of this arrangement.

Theorem 2. Let F_{L}(ℂ, k) be defined as above.

1. The symmetric group Σ_{k} acts on F_{L}(ℂ, k) and the orbit space F_{L}(ℂ, k)/Σ_{k} is homeomorphic to the subspace of monic polynomials of degree k, p(z) ∈ ℂ[z], with the property that the difference of any two roots of p(z), α_{i}, α_{j}, lies outside of the Gaussian integers.

2. It is the complement in ℂ^{k} of the infinite (affine) hyperplane arrangement

A = {H_{i,j}^{σ} | 1 ≤ j < i ≤ k, σ ∈ L}

where H_{i,j}^{σ} = ker(z_{i} − z_{j} − σ).

3. It is an L^{k}-cover of the ordinary configuration space of k points in the torus T = S^{1}×S^{1}. This is a special case of the results in [10], which gives the existence of a principal bundle

L^{k} ⟶ F_{L}(ℂ, k) ⟶ F (T, k).

4. The space F_{L}(ℂ, k) is a K(π, 1).

5. The fibration F_{L}(ℂ, k) → F_{L}(ℂ, k −1) has (i) trivial local coefficients in homology, and (ii) a cross-section.

6. Thus the Lie algebra given by the associated graded for the descending central series of π_{1}(F_{L}(ℂ, k)) is additively isomorphic to the direct sum ⨁_{1<i≤k} L[i] where L[i] is the free Lie algebra generated by elements B_{i,j}^{σ} for fixed i with 1 ≤ j < i ≤ k, and σ runs over the elements of the lattice L.

7. The relations are [B_{i,j}^{σ}, B_{k,i}^{τ}] = [B_{k,i}^{τ}, B_{k,j}^{τ+σ}] [B_{i,j}^{σ}, B_{k,j}^{τ}] = [B_{k,j}^{τ}, B_{k,i}^{τ−σ}]

8. The integral homology of F_{L}(ℂ, k) is additively given by

H_{∗}F_{L}(ℂ, ℓ) ≅ H_{∗}(C_{1}) ⊗ H_{∗}(C_{2}) ⊗ ⋯ ⊗ H_{∗}(C_{k−1})

where C_{i} is the infinite bouquet of circles ⋁_{|Q_{i}^{L}|} S^{1} and Q_{i}^{L} as defined in the beginnig of next section.

Consider the “orbit configuration space” F_{L}(ℂ × ℝ^{n}, k) where L operates diagonally on ℂ × ℝ^{n}, and trivially on ℝ^{n}.

Theorem 3. Assume that q ≥ 1.

1. The loop space ΩF_{L}(ℂ × ℝ^{2q}, k) is homotopy equivalent to the product

∏_{1≤i≤k−1} Ω(ℂ × ℝ^{2q} − Q_{i}^{L})

( although this product decomposition is not multiplicative ).

2. The integral homology of ΩF_{L}(ℂ × R^{2q}, k) is isomorphic to

⨂_{1≤i≤k−1} H_{∗}(Ω(ℂ × ℝ^{2q} − Q_{i}^{L}))

as a coalgebra.

3. The Lie algebra of primitives is isomorphic to the Lie algebra given by π_{∗}(ΩF_{L}(ℂ^{q}, k))/torsion.

4. The Lie algebra of of primitive elements in the homology of ΩF_{L}(ℂ × ℝ^{2q}, k) is a direct sum of free (graded) Lie algebras ⨁_{1<i≤k} L[i] where L[i] is the free graded Lie algebra generated by elements B_{i,j}^{σ} of degree 2q for fixed i with 1 ≤ j < i ≤ k, and σ runs over the elements of the lattice L. The relations are

[B_{i,j}^{σ}, B_{k,i}^{τ}] = [B_{k,i}^{τ}, B_{k,j}^{τ+σ}] [B_{i,j}^{σ}, B_{k,j}^{τ}] = [B_{k,j}^{τ}, B_{k,i}^{τ−σ}]

5. The Lie algebras π_{∗}(ΩF_{L}(ℂ^{q}, k))/torsion, and E_{0}^{∗}(F_{L}(ℂ, k))_{q} are isomorphic as Lie algebras.

2. Proof of Theorem 2

A useful result is the following analogue of the Fadell and Neuwirth fibrations for ordinary configuration spaces. For any natural number ℓ, let Q_{ℓ}^{G} ⊂ M be the union of ℓ distinct orbits Gm_{1}, …, Gm_{ℓ}.

6 F. R. COHEN* AND M. XICOTÉNCATL**

Lemma 4. Let M be a manifold with a properly discontinuous action of a group G and such that the orbit space M/G is again a manifold. Then for ℓ ≤ k, the projection p:F_{G}(M, k) ⟶ F_{G}(M, ℓ) onto the first ℓ coordinates is a locally trivial bundle, with fibre F_{G}(M − Q_{ℓ}^{G}, k − ℓ).

Remark. The extra hypothesis on the action are needed for the case when G is not finite. Notice that they are trivially satisfied when G is a finite group acting freely on M.

Let q_{1}, … q_{k} ∈ ℂ be the points: q_{1} = 0 and q_{i} = q_{i−1} + \frac{1}{2^{i}} for i = 2, …, k. Notice these points lie in different orbits and let Q_{ℓ}^{L} = ⋃_{i=1}^{ℓ}(q_{i} + L) the union of the first ℓ of them. Now an iterated application of this Lemma 4 yields a sequence of of fibrations:

\xymatrix{F_{L}(ℂ, k) \ar[d] & F_{L}(ℂ − Q_{1}^{L}, k − 1) \ar[l] \ar[d] & … \ar[l] & F_{L}(ℂ − Q_{k−1}^{L}, 1) \ar[l] \ar[d]^{≈} \\ ℂ & ℂ − Q_{1}^{L} \ar[u]^{s_{1}} & & ℂ − Q_{k−1}^{L} \ar[u]^{s_{k−1}}} (1)

(where the horizontal maps are the inclusions of the fibers), and sections s_{i}, defined as follows. For z ∈ ℂ set d(z) = min{|z −σ| | σ ∈ L} and notice that d is a continuous function of z. Let y_{1}, …, y_{k−i−1} be k − i − 1 distinct points in the boundary of the disc of radius \frac{1}{16} and centered at the origin. Let

s_{i}:ℂ − Q_{i}^{L} ⟶ F_{L}(ℂ − Q_{i}^{L}, k − i) (2)

be given by:

s_{i}(z) = \begin{cases}(z, d(z)y_{1}, …, d(z)y_{k−i−1}) & if d(z) ≤ \frac{1}{8} \\ (z, y_{1}, …, y_{k−i−1}) & if d(z) ≥ \frac{1}{8} \end{cases}

The local coefficient system for these fibrations is trivial in integral homology. (See Lemma 8 below). Since the homology of the fibre is concentrated in degree one, and the fibration has a cross-section, the Serre spectral sequence for these fibrations collapses. The next proposition follows.

The next proposition gives the additive structure for the integral homology of F_{L}(ℂ, ℓ) as stated in Theorem 2(7).

Proposition 5. The integral homology of F_{L}(ℂ, k) is additively given by

H_{∗}F_{L}(ℂ, ℓ) ≅ H_{∗}(C_{1}) ⊗ H_{∗}(C_{2}) ⊗ ⋯ ⊗ H_{∗}(C_{k−1})

where C_{i} is the infinite bouquet of circles ⋁_{|Q_{i}^{L}|} S^{1}.

In particular, the homology of F_{L}(ℂ, k) is not of finite type and one must proceed with certain care when dualizing. However, something can be said about cohomology.

Proposition 6. The integral cohomology of F_{L}(ℂ, ℓ) is additively given by

H^{∗}F_{L}(ℂ, k) ≅ H^{∗}(C_{1}) ⊗ H^{∗}(C_{2}) ⊗ ⋯ ⊗ H^{∗}(C_{k−1})

Moreover, for every i = 2, … k, there is a choice of cohomology classes in H^{∗}F_{L}(ℂ, k) {A_{i,1}^{σ_{1}}, A_{i,2}^{σ_{2}}, …, A_{i,i−1}^{σ_{i−1}} | σ_{j} ∈ L}, which are in a 1-1 correspondence with the generators in H_{1}(C_{i}), satisfying the following relations:

(a) A_{i,j}^{μ} A_{i,j}^{ν} = 0 for all μ, ν ∈ L.

(b) A_{i,ℓ}^{μ} A_{i,j}^{ν} = A_{j,ℓ}^{μ−ν}(A_{i,j}^{ν} − A_{i,ℓ}^{μ}) if 1 ≤ ℓ < j < i ≤ k and μ, ν ∈ L.

Remark. Perhaps a better way to say this is that H^{∗}F_{L}(ℂ, ℓ) contains a subalgebra isomorphic to the associative, commutative graded algebra A^{∗}, generated by the following collection of degree one elements, {A_{i,j}^{σ}}_{i>j}^{σ∈L}, subject to relations (a) and (b). Moreover, the duals to the standard generators in H_{∗}F_{L}(ℂ, ℓ) realize the the inclusion of algebras A^{∗} ⊂ H^{∗}F_{L}(ℂ, ℓ).

Remark. Notice that in the case μ = ν = 0, relations (a) and (b) recover the relations among the standard generators in H^{∗}F (ℂ, k) (see for example [2]). Roughly speaking, the spaces F (ℂ, k) and F_{L}(ℂ, k) have locally the same cohomology.

8 F. R. COHEN* AND M. XICOTÉNCATL**

Proof. Begin by identifying the classes A_{i,j}^{σ}. In the case k = 2, the projection onto the first coordinate gives a fibration: (ℂ − Q_{1}^{L}) ⟶ F_{L}(ℂ, k) ⟶ ℂ for which the inclusion of the fiber over q_{1} = 0, is a homotopy equivalence. Therefore:

F_{L}(ℂ, 2) ≃ ⋁_{L} S^{1}

For every σ ∈ L let p_{σ}:F_{L}(ℂ, 2) ⟶ S^{1} be the map

p_{σ}(x, y) = \frac{y − x − σ}{|y − x − σ|}

and put A_{σ} = p_{σ}^{∗}(i) where i is a fixed generator of H^{1}(S^{1}) = ℤ. Then {A_{σ}}_{σ∈L} is a linearly independent family in H^{1}F_{L}(ℂ, 2), where each A_{σ} corresponds to one of the circles in the bouquet.

Finally, let π_{i,j}:F_{L}(ℂ, k) ⟶ F_{L}(ℂ, 2) be the projections given by π_{i,j}(x_{1}, …, x_{k}) = (x_{j}, x_{i}) for j < i. Define A_{i,j}^{σ} = π_{i,j}^{∗}(A_{σ}) for σ ∈ L. Thus A_{i,j}^{σ}A_{i,j}^{τ} for all σ, τ as A_{σ}A_{τ} = 0 in H^{∗}F_{L}(ℂ, 2) ≅ H^{∗}(⋁_{L} S^{1}).

To check relation (b), it suffices to check the case when i = 3, j = 2, ℓ = 1 by applying the projections π_{i,j,ℓ}: F_{L}(ℂ, k) ⟶ F_{L}(ℂ, 3), together with naturality. The cohomology classes A_{2,1}^{σ}, A_{3,1}^{σ},A_{3,2}^{σ} are the duals to the homology classes represented by the maps: φ_{i,j}^{σ}: S^{1} ⟶ F_{L}(ℂ, 3)

φ_{2,1}^{σ}(z) = (q_{1}, q_{1} + σ + \frac{z}{8}, q_{3})
φ_{3,1}^{σ}(z) = (q_{1}, q_{2}, q_{1} + σ + \frac{z}{8})
φ_{3,2}^{σ}(z) = (q_{1}, q_{2}, q_{2} + σ + \frac{z}{16})

which generate H_{1}(C_{1}) and H_{1}(C_{2}). From the sequence of fibrations (1) for k = 3, it follows that H^{∗}F_{L}(ℂ, 3) ≅ H^{∗}(C_{1}) ⊗ H^{∗}(C_{2}) and therefore

A_{i,j}^{μ}A_{s,t}^{ν} = ∑_{σ,τ} α_{σ,τ} A_{2,1}^{σ}A_{3,1}^{τ} + ∑_{σ,τ} β_{σ,τ} A_{2,1}^{σ}A_{3,2}^{τ}

where α_{σ,τ} and β_{σ,τ} are integers to be determined. For this purpose there are maps which detect the A_{2,1}^{σ}A_{3,1}^{τ} and A_{2,1}^{σ}A_{3,2}^{τ} and then study their effect on A_{i,j}^{μ}A_{s,t}^{ν}. For fixed σ, τ ∈ L, let F_{1}, F_{2} :S^{1} × S^{1} ⟶ F_{L}(ℂ, 3) be the maps

F_{1}(z, w) = (q_{1}, q_{1} + σ + \frac{z}{8}, q_{1} + τ + \frac{w}{16})
F_{2}(z, w) = (q_{1}, q_{1} + σ + \frac{z}{8}, q_{1} + σ + τ + \frac{z}{8} + \frac{w}{16}).

It is immediate to verify their effect in cohomology:

(F_{1})^{∗}(A_{2,1}^{λ}) = \begin{cases}ι ⊗ 1 & if λ = σ \\ 0 & otherwise \end{cases}

(F_{1})^{∗}(A_{3,1}^{λ}) = \begin{cases}1 ⊗ ι & if λ = τ \\ 0 & otherwise \end{cases}

(F_{1})^{∗}(A_{3,2}^{λ}) = \begin{cases}ι ⊗ 1 & if λ = τ − σ \\ 0 & otherwise \end{cases}

(F_{2})^{∗}(A_{2,1}^{λ}) = \begin{cases}ι ⊗ 1 & if λ = σ \\ 0 & otherwise \end{cases}

(F_{2})^{∗}(A_{3,1}^{λ}) = \begin{cases}ι ⊗ 1 & if λ = σ + τ \\ 0 & otherwise \end{cases}

(F_{2})^{∗}(A_{3,2}^{λ}) = \begin{cases}1 ⊗ ι & if λ = τ \\ 0 & otherwise \end{cases}

Thus, the homomorphisms F_{1}^{∗} and F_{2}^{∗} detect the basic products A_{2,1}^{σ}A_{3,1}^{τ} and A_{2,1}^{σ}A_{3,2}^{τ}, respectively. Since they are also ring homomorphisms, we have

(F_{1})^{∗}(A_{3,1}^{μ}A_{3,2}^{ν}) = \begin{cases}−ι ⊗ ι & if μ = τ, ν = τ − σ \\ 0 & otherwise \end{cases}

(F_{2})^{∗}(A_{3,1}^{μ}A_{3,2}^{ν}) = \begin{cases}ι ⊗ ι & if μ = σ + τ, ν = τ \\ 0 & otherwise \end{cases}

and therefore we get: A_{3,1}^{μ}A_{3,2}^{ν} = −A_{2,1}^{μ−ν}A_{3,1}^{μ} + A_{2,1}^{μ−ν}A_{3,2}^{ν}. □

Lemma 7. Let p : E → B be a locally trivial bundle with B path-connected. Let x, y ∈ B and let F_{x} and F_{y} be the corresponding fibers. Then π_{1}(B, x) acts trivially on H_{∗}(F_{x}) if and only if π_{1}(B, y) acts trivially on H_{∗}(F_{y}).

Proof. Assume that π_{1}(B, y) acts trivially on H_{∗}(F_{y}). Let α : I → B be a path from x to y and let

α̂ : π_{1}(B, y) ⟶ π_{1}(B, x)
[γ] ↦ [α] ∗ [γ] ∗ [α]^{−1}

be the induced isomorphism of fundamental groups. We prove that α̂[γ] acts trivially on H_{∗}(F_{x}). Let:

• G be a lifting for γ.

• A be a lifting for α and

• Ā be a lifting for ᾱ, where ᾱ(t) = α(1 − t). This is, G, A and Ā are maps such that the following diagrams commute:

\xymatrix{I × F_{y} \ar[r]^{G} \ar[d] & E \ar[d]^{p} & I × F_{x} \ar[r]^{A} \ar[d] & E \ar[d]^{p} & I × F_{y} \ar[r]^{Ā} \ar[d] & E \ar[d]^{p} \\ I \ar[r]^{γ} & B & I \ar[r]^{α} & B & I \ar[r]^{ᾱ} & B}

To construct a lifting for α ∗ γ ∗ ᾱ, define H : I × F_{x} → E by

H(t, e) = \begin{cases}A(3t, e) & if 0 ≤ t ≤ \frac{1}{3} \\ G(3t − 1, A(1, e)) & if \frac{1}{3} ≤ t ≤ \frac{2}{3} \\ Ā(3t − 2, G(1, A(1, e))) & if \frac{1}{2} ≤ t ≤ 1 \end{cases}

Notice that in this case, the right end of the homotopy H_{1} = H |_{t=1} is a continuous mapping F_{x} → F_{x} given by H_{1}(e) = Ā(1, G(1, A(1, e))), that is to say H_{1} = Ā_{1} ∘ G_{1} ∘ A_{1} and this composite induces a commutative diagram of homology groups:

\xymatrix{H_{∗}(F_{x}) \ar[r]^{(H_{1})_{∗}} \ar[d]_{(A_{1})_{∗}} & H_{∗}(F_{x}) \\ H_{∗}(F_{y}) \ar[r]^{(G_{1})_{∗}} & H_{∗}(F_{y}) \ar[u]_{(Ā_{1})_{∗}}}

where (G_{1})_{∗} = 1_{H_{∗}(F_{y})} since [γ] acts trivially on H_{∗}(F_{y}). Thus, (H_{1})_{∗} = (Ā_{1})_{∗} ∘ (A_{1})_{∗}. But this is precisely the isomorphism of H_{∗}(F_{x}) induced by the trivial loop α ∗ ᾱ and then (H_{1})_{∗} must be the identity. Therefore, α̂[γ] acts trivially on H_{∗}(F_{x}). □

Remark. The same proof works if using cohomology instead of homology. The way in which Lemma 7 is actually used is the following. Consider the fiber bundle given by projection onto the first coordinate π: F_{L}(ℂ − Q_{r}^{L}, k − r) ⟶ (ℂ − Q_{r}^{L}) . Let z_{0} ∈ ℂ − Q_{r}^{L} be a base point and then

π_{1}(ℂ − Q_{r}^{L}, z_{0}) ≅ F [x_{1}^{σ_{1}}, …, x_{r}^{σ_{r}} | σ_{i} ∈ L]

One uses the that for each generator x_{i}^{σ_{i}}, there exist a point z_{i,σ_{i}} ∈ ℂ − Q_{r}^{L} (namely take z_{i,σ_{i}} = q_{r+1} + σ_{i}) for which is relatively easy to prove that the action of the corresponding generator in π_{1}(ℂ − Q_{r}^{L}, z_{i,σ_{i}}) is trivial on the homology of the fiber over z_{i,σ_{i}}. Since ℂ − Q_{r}^{L} is path connected, Lemma 7 implies that x_{i}^{σ_{i}} acts trivially on the homology of the fiber over z_{0}.

Lemma 8. The fibration π:F_{L}(ℂ − Q_{r}^{L}, k − r) ⟶ ℂ − Q_{r}^{L} has trivial local coefficients.

12 F. R. COHEN* AND M. XICOTÉNCATL**

Proof. The proof is by downward induction on r and it is the equivariant analogue to the proof given in [4]. For r = k − 1 the result is clear. Assume the result for r + 1. Consider the fibration π:F_{L}(ℂ − Q_{r}^{L}, k − r) ⟶ ℂ − Q_{r}^{L} with fibre F_{L}(ℂ − Q_{r+1}^{L}, k − r − 1). Define a function

ρ_{i}:I × (ℂ − Q_{r}^{L}) ⟶ (ℂ − Q_{r}^{L})

as follows. Let D_{i} be a disc containing the points q_{i} and q_{r+1} as in [4] pg. 252, with two additional conditions:

[figure: • • •  • • •  • • • D_{i}  D_{i} + (1, 1) •q_{i} •q_{r}  •q_{r+1}  … ⋮  q_{1} + (1, 1) •q_{1} …; x; y]

(i) D_{i} must be entirely contained in the interior of a unit square, so that D_{i} ∩(D_{i} +σ) = ∅ for σ ≠ 0.

(ii) D_{i} ∩ B(0, \frac{1}{8}) = ∅ for i ≠ 1, and D_{1} ∩ B(q_{2}, \frac{1}{8}) = ∅, where B(z, r) denotes the closed ball in with center z and radius r. Since q_{i} = 0+ \frac{1}{4} +⋯+ \frac{1}{2^{i}}, this is not difficult to achieve. The function ρ_{i} does the following:

ORBIT CONFIGURATION SPACES ... 13

• Inside D_{i} rotates the 2-disc with center q_{i} cointained within the 2-annulus by an angle of 2πt at time t, fixes the boundary of D_{i} and appropriately “twists” the 2-annulus, at time t to insure the continuity of ρ_{i}.

• Twists all discs (D_{i} + σ), for σ ∈ L, in such a way that the resulting function is L-equivariant, namely

ρ_{i}(t, z) = ρ_{i}(t, z − σ) + σ ∀z ∈ (D_{i} + σ).

• Fixes the complement of the union of all discs ℂ − ⋃_{σ∈L}(D_{i} + σ). Recall that π_{1}(ℂ − Q_{r}^{L}) is isomorphic to the free group F [ x_{i}^{σ} | 1 ≤ i ≤ r, σ ∈ L]. Using the function ρ_{i}, we prove that all generators x_{i}^{σ} act trivially on H_{∗}F_{L}(ℂ − Q_{r+1}^{L}, k − r − 1). More precisely, it will follow that the generator corresponding to the point q_{i} + σ acts trivially on the homology of the fiber over q_{r+1} + σ. By Lemma 7, this is enough to insure the triviality of the local coefficient system.

Define γ_{i,σ}(t) = ρ_{i}(t, q_{r+1} + σ). Then γ_{i,σ} : I → (ℂ − Q_{r}^{L}) is a generator of the group π_{1}(ℂ − Q_{r}^{L}, q_{r+1} + σ) which corresponds to the point q_{i} + σ. Define a “lift”

H_{i,σ} : I × F_{L}(ℂ − Q_{r+1}^{L}, k − r − 1) ⟶ F_{L}(ℂ − Q_{r}^{L}, k − r)

of γ_{i,σ} by:

H_{i,σ}(t; z_{1}, z_{2}, …, z_{k−r−1}) = (ρ_{i}(t, q_{r+1} + σ), ρ_{i}(t, z_{1}), …, ρ_{i}(t, z_{k−r−1}),) (3)

Notice that the right side of (3) lies on F_{L}(ℂ − Q_{r}^{L}, k − r) since z_{j} ≢ q_{r+1} mod L, z_{j} ≢ z_{ℓ} mod L and ρ_{i} is an L-equivariant isotopy. Then we have a commutative diagram:

\xymatrix{I × F_{L}(ℂ − Q_{r+1}^{L}, k − r − 1) \ar[d] \ar[r]^{H_{i,σ}} & F_{L}(ℂ − Q_{r}^{L}, k − r) \ar[d] \\ I \ar[r]^{γ_{i,σ}} & ℂ − Q_{r}^{L}}

By definition, the local coefficient system is trivial if the map

(H_{i,σ})_{1} :F_{L}(ℂ − Q_{r+1}^{L}, k − r − 1) ⟶ F_{L}(ℂ − Q_{r+1}^{L}, k − r − 1)

is the identity in homology. Let B_{i,j}^{σ} be the homology dual of A_{i,j}^{σ}. Recall that

H_{∗}F_{L}(ℂ − Q_{r+1}^{L}, k − r − 1) ≅ ⨂_{j=2}^{k−r} ⟨ B_{r+j,1}^{σ_{1}}, …, B_{r+j,r+j−1}^{σ_{r+j−1}} ⟩ (4)

so one must check the effect of (H_{i,σ})_{1}_{∗} on the B_{r+j,ℓ}^{σ} ’s. For a fixed j, 1 ≤ j ≤ k − r − 1, there are sections

\xymatrix{F_{L}(ℂ − Q_{r+j}^{L}, k − r − j) \ar[r] & ℂ − Q_{r+j}^{L} \ar[l]^{s_{r+j}}}

such that the classes B_{r+j+1,ℓ}^{σ} for ℓ = 1, …, (r + j) and σ ∈ L, are the images in homology of the generators H_{1}(ℂ − Q_{r+j}^{L}) under s_{r+j}, followed by the morphism induced by the inclusion in F_{L}(ℂ − Q_{r+1}^{L}, k − r − 1). So, to prove that [γ_{i,σ}] acts trivially on the j-th factor of (4), notice that the map (H_{i,σ})_{1} induces maps of fibrations

\xymatrix{F_{L}(ℂ − Q_{r+j}^{L}, k − r − j) \ar[r]^{Ĥ_{i,σ}} \ar[d] & F_{L}(ℂ − Q_{r+j}^{L}, k − r − j) \ar[d] \\ ℂ − Q_{r+j}^{L} \ar[u]^{s_{r+j}} \ar[r]^{H̄_{i,σ}} & ℂ − Q_{r+j}^{L} \ar[u]^{s_{r+j}}}

which are compatible with the sections, in the sense that Ĥ_{i,σ} ∘ s_{r+j} = s_{r+j} ∘ H̄_{i,σ}

For i = 2, …, r, the sections given by (2) suffice. For i = 1, there is a straighforward modification of s_{1} that uses B(q_{2}, \frac{1}{8}) instead of B(0, \frac{1}{8}). Then it will be enough to show that

(H̄_{i,σ})_{∗}:H_{1}(ℂ − Q_{r+j−1}^{L}) ⟶ H_{1}(ℂ − Q_{r+j−1}^{L})

is the identity morphism and this is clear, since at the level of fundamental groups

(H̄_{i,σ})_{♯}:π_{1}(ℂ − Q_{r+j}^{L}) ⟶ π_{1}(ℂ − Q_{r+j}^{L})

the induced homomorphism is given by either: the identity, conjugation by a fixed element in π_{1} or multiplicacion by a certain commutator, see [4] pg. 254. □

ORBIT CONFIGURATION SPACES ... 15

3. Proof of Theorem 1

Recall that a multiplicative fibration with section is homotopy equivalent to a product. Thus ΩX(ℝ^{n}, k) is homotopy equivalent to ΩX(ℝ^{n}, k − 1) × Ω(ℝ^{n} − S_{k−1}), and the first part of the theorem follows by induction.

The second part of the theorem follows from the Künneth theorem, and part 1 of the theorem.

Since ℝ^{n} −S_{k−1} has the homotopy type of a (possibly infinite ) bouquet of (n−1)-spheres, the homology of its loop space follows from the Bott-Samelson theorem. In this case, it is well-known that there are isomorphisms of Lie algebras

(5) π_{∗}(Ω(ℝ^{n} − S_{k−1})/torsion → P rimH_{∗}(Ω(ℝ^{n} − S_{k−1}).

Furthermore, the existence of sections implies that the Hurewicz homomorphism

π_{∗}(ΩX(ℝ^{n}, k) → P rimH_{∗}(ΩX(ℝ^{n}, k)

is a surjection. Since the map (5) is an injection, the theorem follows.

4. Proof of Theorem 3

Consider the fibration with section F_{L}(ℂ × ℝ^{2q}, k) → F_{L}(ℂ × ℝ^{2q}, k − 1). The fibre of this map is ℂ × ℝ^{2q} − Q_{k−1}^{L}. By Theorem 1, the conclusions of Theorem 3 all follow except possibly the last two which state the precise extension of Lie algebras.

To finish, it suffices to prove parts (4), and (5) of the theorem by a direct comparison of the two Lie algebras π_{∗}(ΩF_{L}(ℂ × ℝ^{2q}, k))/torsion, and E_{0}^{∗}(F_{L}(ℂ, k))_{q}. The computation of the Lie algebra attached to the descending central series, is just an explicit a calculation that involves the local coefficient system and the generators constructed above.

Thus define maps F_{i} : S^{2q+1} × S^{2q+1} → F_{L}(ℂ × ℝ^{2q}, 3) analogous to those in the proof of the relations for the cohomology of F_{L}(ℂ, k) by the formulas:

F_{1}(z, w) = (q_{1}, q_{1} + σ + \frac{z}{8}, q_{1} + τ + \frac{w}{16})
F_{2}(z, w) = (q_{1}, q_{1} + σ + \frac{z}{8}, q_{1} + σ + τ + \frac{z}{8} + \frac{w}{16}).

Consider the loopings of these maps Ω(F_{i}) : Ω(S^{2q+1} × S^{2q+1}) → Ω(F_{L}(ℂ × ℝ^{2q}, 3)). Notice that the fundamental cycles in degree 2q for the integer homology of Ω(S^{2q+1} ×S^{2q+1}) commute. Thus it suffices to calculate the image of the fundamental cycles in the homology of Ω(F_{L}(ℂ × ℝ^{2q}, 3)). This gives the precise relations as stated in parts (4)-(5) of Theorem 3 which follows at once.

Acknowledgements

We would like to thank the organizers of “Arrangements in Boston” for the opportunity to participate on this very interesting, and stimulating conference which informed on many parts of mathematics. This paper was written while the second author was a visitor at the Max-Planck-Institut für Mathematik. He would like to thank the Institute for its financial support, hospitality and for providing such a great working enviroment.

References

[1] D. C. Cohen, Monodromy of fibre-type arrangements and orbit configuration spaces, to appear in Forum Mathematicum.

[2] F.R. Cohen, On configuration spaces, their homology, and Lie Algebras, Journal of Pure and Applied Algebra 100 (1995), 19-42.

[3] F.R. Cohen and S. Gitler, Loop spaces of configuration spaces, braid-like groups, and knots, To appear in in the Proceedings of the 1998 Barcelona Conference in Algebraic Topology.

[4] F.R. Cohen, T.J. Lada and J.P. May, The homology of iterated loop spaces, Lecture Notes in Math., vol. 533, Springer-Verlag (1976).

[5] E. Fadell and S. Husseini, The space of loops on configuration spaces and the Majer-Terracini index, Topol. Methods Nonlinear Anal. 11 (1998), 249–271.

[6] E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand., 10 (1962), 111-118.

ORBIT CONFIGURATION SPACES ... 17

[7] T. Kohno, Linear represenations of braid groups and classical Yang-Baxter equations, Cont. Math. 78(1988), 339-363.

[8] T. Kohno, Vassiliev invariants and de Rham complex on the space of knots, Cont. Math. 179(1994), 123-138.

[9] T. Kohno, Elliptic KZ system, braid groups of the torus and Vassiliev invariants, Topology and its Applications, 78(1997), 79-94.

[10] M.A. Xicoténcatl, Ph.D. Thesis, The University of Rochester (1997).

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

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

Max-Planck-Institut für Mathematik, P.O. Box 7280, D-53072 Bonn, Germany. Depto. de Matemáticas, Cinvestav, Apdo. Postal 14-740, México, D.F. 07300.

E-mail address: xico@mpim-bonn.mpg.de, xico@math.cinvestav.mx