Product Decomposition of Loop Spaces of Configuration Spaces.

Miguel A. Xicoténcatl

Max-Planck-Institut für Mathematik, (current address) P.O. Box 7280, D-53072 Bonn, Germany.

xico@mpim-bonn.mpg.de

Departamento de Matemáticas, CINVESTAV - IPN Apdo. Postal 14-740 México, D.F. 07300, México.

xico@math.cinvestav.mx

Abstract The configuration space of k points in ℝP^{n}, ℂP^{n} and ℍP^{n} are studied. In this article we show that after looping once, they split as a product of spheres and the loop space of certain orbit configuration spaces.

1991 Mathematics Subject Classification: 55P35, 55R10,55R80, 57S25. Keywords and phrases: Loop spaces of configuration spaces, Orbit configuration spaces.

1 Introduction and main result

Let M be a topological space and G a group acting freely on M. Let Gm denote the orbit of an element m ∈ M under the action of G. Define the orbit configuration space of k points in M by

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

Such spaces were introduced in [6] as generalizations of the ordinary configuration spaces defined by Fadell and Neuwirth [5]. Indeed, if G is the trivial group, the space F_{G}(M, k) coincides with the usual configuration space F (M, k) = { (m_{1}, …, m_{k}) ∈ M^{k} | m_{i} ≠ m_{j} for i ≠ j }. The purpouse of this note is to exhibit product decompositions of the loop space of F (M, k) for M = ℝP^{n}, ℂP^{n} and ℍP^{n}, in terms of loop spaces of orbit configuration spaces and known spaces.

An equivalent description of F_{G}(M, k) can be given in terms of ordinary configuration spaces. Let f:M/G → BG be the map which classifies the pricipal bundle G → M → M/G. The following theorem was proven in [7]:

Theorem 1.1 The space F_{G}(M, k) is homeomorphic to the total space of the pull-back of the principal fibration G^{k} ⟶ (EG)^{k} ⟶ (BG)^{k} along the composition F (M/G, k) ↪

(M/G)^{k} ⟶^{f^{k}} (BG)^{k}.

Therefore, there is a principal G^{k}-bundle, F_{G}(M, k) → F (M/G, k). More precisely, the group G^{k} acts coordinatewise on the space F_{G}(M, k) and the quotient F_{G}(M, k)/G^{k} is homeomorphic to F (M/G, k). In particular there is a fibration (up to homotopy):

F_{G}(M, k) ⟶ F (M/G, k) ⟶ BG^{k}  (1)

For the rest of the article, we will only be concerned with the following special cases:

(i) the ℤ_{2} action on S^{n} given by the antipodal map,

(ii) the S^{1} action on S^{2n+1} given by complex multiplication and

(iii) the S^{3} action on S^{4n+3} given by quaternionic multiplication,

having as orbit spaces the projective spaces ℝP^{n}, ℂP^{n} and ℍP^{n}, respectively. It will be shown in section 2 that in cases (i), (ii) and (iii), the corresponding fibrations (1) admit a section after looping and they split as products. Thus the main result can be stated as follows:

Theorem 1.2 For every n, k ≥ 1 there are homotopy equivalences:

(a) ΩF (ℝP^{n}, k) ≃ (ℤ_{2})^{k} × ΩF_{ℤ_{2}}(S^{n}, k)  if n ≥ 3
(b) ΩF (ℂP^{n}, k) ≃ (S^{1})^{k} × ΩF_{S^{1}}(S^{2n+1}, k)  if n ≥ 2
(c) ΩF (ℍP^{n}, k) ≃ (S^{3})^{k} × ΩF_{S^{3}}(S^{4n+3}, k)  if n ≥ 2

It is important to mention here that loop spaces of configuration spaces have received a lot of attention recently. This is due to some interesting relations between their homology and the Vassiliev invariants of braids [3], the Lie algebra associated to the descending central series of braid groups, [3], [6] and the n-body problem [4], among other topics.

2 Some auxiliary lemmas

The proof of Therorem 1.2 is given after the following lemmas:

Lemma 2.1 For n ≥ 3, the natural inclusion \xymatrix{F (ℝP^{n}, k) \ar[r] & (ℝP^{∞})^{k}} is an isomorphism on fundamental groups.

Proof. The statement of the lemma is clear for k = 1. By induction, assume this is true for k − 1 and consider the map of fibrations induced by the inclusion

\xymatrix{(ℝP^{n} − Q_{k−1}) \ar[r] \ar[d] & ℝP^{∞} \ar[d] \\ F (ℝP^{n}, k) \ar[r] \ar[d] & (ℝP^{∞})^{k} \ar[d] \\ F (ℝP^{n}, k − 1) \ar[r] & (ℝP^{∞})^{k−1}}

where both fibrations are projection onto the first k − 1 coordinates, and Q_{k−1} denotes a set of k − 1 distinct points in ℝP^{n}. Notice that (ℝP^{n} − Q_{k−1}) homotopy equivalent to ℝP^{n−1} ∨ (⋁_{k−2} S^{n−1}) and so the natural inclusion (ℝP^{n} − Q_{k−1}) ⟶ ℝP^{∞} gives an isomorphism of fundamental groups (both isomorphic to ℤ_{2}). By induction hypothesis, the map between the base spaces also induces an isomorphism on π_{1}. The lemma follows. □

Similarly we have:

Lemma 2.2 For n ≥ 2, the natural inclusion \xymatrix{F (ℂP^{n}, k) \ar[r] & (ℂP^{∞})^{k}} is an isomorphism on π_{2}.

Lemma 2.3 For n ≥ 2, the natural inclusion \xymatrix{F (ℍP^{n}, k) \ar[r] & (ℍP^{∞})^{k}} is an isomorphism on π_{4}.

Thus after looping the fibration F_{ℤ_{2}}(S^{n}, k) ⟶ F (ℝP^{n}, k) ⟶ (Bℤ_{2})^{k} we obtain a principal fibration ΩF_{ℤ_{2}}(S^{n}, k) ⟶ ΩF (ℝP^{n}, k) ⟶ (ℤ_{2})^{k} where the projection induces an isomorphism on π_{0}. Therefore ΩF (ℝP^{n}, k) is indeed the disjoint union of 2^{k} copies of ΩF_{ℤ_{2}}(S^{n}, k). Thus we have ΩF (ℝP^{n}, k) ≃ (ℤ_{2})^{k} × ΩF_{ℤ_{2}}(S^{n}, k). Part (a) follows.

For case (b), consider the fibration: F_{S^{1}}(S^{2n+1}, k) ⟶ F (ℂP^{n}, k) ⟶ (BS^{1})^{k}. Since the projection induces an isomorphism on π_{2}, then after looping, the projection induces an isomorphism on π_{1} and there is a lifting

\xymatrix{ & ΩF (ℂP^{n}, k) \ar[d] \\ (⋁_{k} S^{1}) \ar@{-->}[ur] \ar[r] & Ω(BS^{1})^{k}.}

Thus by Lemma 2.4 below there exists a section for the multiplicative fibration ΩF_{ℤ_{2}}(S^{2n+1}, k) ⟶ ΩF (ℂP^{n}, k) ⟶ (S^{1})^{k} and part (b) follows.

For case (c), consider the fibration: F_{S^{3}}(S^{4n+3}, k) ⟶ F (ℍP^{n}, k) ⟶ (BS^{3})^{k}. Since the projection induces an isomorphism on π_{4}, then after looping, the projection induces an isomorphism on π_{3} and there is a lifting

\xymatrix{ & ΩF (ℍP^{n}, k) \ar[d] \\ (⋁_{k} S^{3}) \ar@{-->}[ur] \ar[r] & Ω(BS^{3})^{k}.}

Thus by Lemma 2.4 there exists a section for the multiplicative fibration ΩF_{ℤ_{2}}(S^{4n+3}k) ⟶ ΩF (ℍP^{n}, k) ⟶ (S^{3})^{k} and part (c) follows.

We prove now the lemma needed in cases (b) and (c). Let i denote the inclusion i : ∨_{k}G ↪ G^{k} = Ω(BG)^{k}.

Lemma 2.4 Let α : Z ⟶ (BG)^{k} be a fibration. Assume that after looping, the map i : ∨_{k}G ⟶ Ω(BG)^{k} admits a homotopy lifting to ΩZ that is, there is a map f:∨_{k}G → ΩZ such that Ωα ∘ f ≃ i. Then, f extends to a homotopy section, that is there is a map F:G^{k} → ΩZ such that the following diagram homotopy commutes

\xymatrix{G^{k} \ar@{-->}[r]^{F} & ΩZ \ar[d]^{Ωα} \\ ∨_{k}G \ar@{^{(}->}[u] \ar[ur]^{f} \ar[r]^{i} & Ω(BG)^{k}}

and the composite: \xymatrix{G^{k} \ar[r]^{F} & ΩZ \ar[r]^{Ωα} & Ω(BG)^{k}} is homotopic to the identity.

We prove the lemma in the case k = 2. The proof for general k is similar and requires the generalization of the statements below to k factors. As preparation we need the following facts. Spaces are assumed to be in T_{∗}, the category of compactly generated, weak Hausdorff spaces, with non-degenerate base points.

Recall from [1] that there are maps

f_{i} : X × Y ⟶ ΩΣ(X ∨ Y) i = 1, 2, 3

given as follows. The maps f_{1} and f_{2} are given by the compositions:

\xymatrix{X × Y \ar[r]^{proj} & X \ar[r]^{σ_{X}} & ΩΣX \ar[r]^{j_{1}} & ΩΣ(X ∨ Y) \\ X × Y \ar[r]^{proj} & Y \ar[r]^{σ_{Y}} & ΩΣY \ar[r]^{j_{2}} & ΩΣ(X ∨ Y)}

where σ_{X} (respectively σ_{Y}) is the adjoint to the identity map ΣX → ΣX (respectively ΣY → ΣY), j_{1}, j_{2} are the natural inclusions and f_{3} is given by the Samelson product map:

X × Y ⟶ X ∧ Y ⟶^{i_{X}∧i_{Y}} (X ∨ Y) ∧ (X ∨ Y) ⟶^{[ , ]} ΩΣ(X ∨ Y)

The difference γ = f_{1} + f_{2} − f_{3} in the group [X × Y, ΩΣ(X ∨ Y)] satisfies the following properties:

(1) When restricted to X ∨Y, γ is homotopic to the suspension σ : X ∨Y ⟶ ΩΣ(X ∨Y)

(2) The map γ composed with the loops on the projection ΩΣ(X ∨ Y) ⟶ ΩΣX is homotopic to the map gotten as the composite

\xymatrix{X × Y \ar[r]^{proj} & X \ar[r]^{σ_{X}} & ΩΣX}

(3) The map γ composed with the loops on the projection ΩΣ(X ∨ Y) ⟶ ΩΣY is homotopic to the map gotten as the composite

\xymatrix{X × Y \ar[r]^{proj} & Y \ar[r]^{σ_{Y}} & ΩΣY}

The next Lemma follows at once:

Lemma 2.5 The map γ considered above is an extension of the suspension σ : X ∨ Y ⟶ ΩΣ(X ∨ Y) over the product X × Y. Furthermore, the composite

\xymatrix{X × Y \ar[r]^{γ} & ΩΣ(X ∨ Y) \ar[r]^{Ω(θ)} & ΩΣX × ΩΣY}

is homotopic to the product σ_{X} × σ_{Y}, where θ : ΣX ∨ ΣY ⟶ ΣX × ΣY is the natural inclusion.

Lemma 2.6 Let α : A ⟶ Σ(X ∨ Y) which admits a section. Taking adjoints this implies that the suspension σ : X ∨ Y ⟶ ΩΣ(X ∨ Y) admits a homotopy lifting

\xymatrix{ & Ω(A) \ar[d]^{Ω(α)} \\ X ∨ Y \ar@{-->}[ur] \ar[r]^{σ} & ΩΣ(X ∨ Y)}

Then there exists an extension X × Y ⟶ Ω(A) such that the composite:

\xymatrix{X × Y \ar[r] & Ω(A) \ar[r] & ΩΣ(X ∨ Y) \ar[r] & ΩΣX × ΩΣY}

is homotopic to σ_{X} × σ_{Y}.

Proof. We indicate the construction of a lifting. The map X ∨ Y ⟶ Ω(A) can be expressed as g_{1} ∨ g_{2} : X ∨ Y ⟶ Ω(A) where g_{1} : X → Ω(A) and g_{2} : Y → Ω(A). Then the following diagram is homotopy commutative:

\xymatrix{X × Y \ar[dr] \ar[r]^{g_{1}×g_{2}} & Ω(A) × Ω(A) \ar[d] \ar[r]^{mult} & Ω(A) \ar[d] \\  & ΩΣ(X ∨ Y)^{2} \ar[r] & ΩΣ(X ∨ Y) \ar[r] & ΩΣX × ΩΣY}

The desired map is given by the composite on the top. The second assertion follows now from the previous lemma. □

Proof (of Lemma 2.4). Let α : Z ⟶ BG × BH and assume that after looping the inclusion G ∨ H → Ω(BG × BH) admits a lifting G ∨ H → ΩZ. If we take X = G and Y = H in the previous lemmas, we get a homotopy commutative diagram:

[figure: G × H; γ; ΩΣ(G ∨ H); lift; ΩZ; Ω(α); G ∨ H; ΩΣ(G ∨ H); Ω(BG ∨ BH); Ω(BG × BH)]

where

1. The left hand square homotopy commutes by Lemma 2.5.

2. The right hand square homotopy commutes by hypothesis.

3. The composite:

\xymatrix{G × H \ar[r] & ΩΣ(BG ∨ BH) \ar[r] & ΩΣG × ΩΣH \ar[r] & ΩBG × ΩBH}

is a homotopy equivalence. This a direct consequence of the fact that the natural composite G ⟶ ΩΣG ⟶ ΩBG is a homotopy equivalence.

This gives the desired section and the lemma follows. □

References

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[2] F.R. Cohen, A course in some aspects of classical homotopy theory, in Algebraic topology proceedings, Seattle 1985, Lecture Notes in Math., vol. 1286, Springer-Verlag (1987).

[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] 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.

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

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

[7] M.A. Xicoténcatl, On configuration spaces of orbits of points and their loop space homology, Submitted to Transactions of the AMS.