Homology, Homotopy and Applications, vol.6(1), 2004, pp.167–173

HOMOTOPY LIE ALGEBRA OF CLASSIFYING SPACES FOR HYPERBOLIC COFORMAL 2-CONES

J.-B. GATSINZI

(communicated by Johannes Huebschmann)

Abstract In this paper, we show that the rational homotopy Lie algebra of classifying spaces for certain types of hyperbolic coformal 2-cones is not nilpotent.

1. Introduction

A simply connected space X is called an n-cone if it is built up by a sequence of cofibrations

Y_{k} →^{f} X_{k−1} →^{j_{k}} X_{k}

with X_{0} = ∗ and X_{n} ≃ X. One can further assume that Y_{k} ≃ Σ^{k−1}W_{k} is a (k − 1)- fold suspension of a connected space W_{k} [3]. In particular a 2-cone X is the cofibre of a map between two suspensions

ΣA →^{f} ΣB → X.  (1)

Spaces under consideration are assumed to be 1-connected and of finite type, that is, H^{i}(X; ℚ) is a finite-dimensional ℚ-vector space. To every space X corresponds a free chain Lie algebra of the form (𝕃(V), δ) [2], called a Quillen model of X. It is an algebraic model of the rational homotopy type of X. In particular, one has an isomorphism of Lie algebras H_{∗}(𝕃(V), δ) ≅ π_{∗}(ΩX) ⊗ ℚ. The model is called minimal if δV ⊂ 𝕃^{⩾2}(V). A space X is called coformal if there is a map of differential Lie algebras (𝕃(V), δ) → (π_{∗}(ΩX)⊗ℚ, 0) that induces an isomorphism in homology. Any continuous map f : X → Y has a Lie representative f̃: (𝕃(W), δ′) → (𝕃(V), δ) between respective models of X and Y.

If X is a 2-cone as defined by (1) and f̃: 𝕃(W) → 𝕃(V) is a model of f, then a Quillen model of the cofibre X of f is obtained as the push out of the following diagram:

[figure: (𝕃(W), 0); f̃; ı; (𝕃(V), 0); ī; (𝕃(W ⊕ sW), d); f̄; (𝕃(V ⊕ sW), δ)]

This work was supported by the Abdus Salam ICTP in cooperation with SIDA. Received November 14, 2003, revised April 21, 2004; published on May 3, 2004. 2000 Mathematics Subject Classification: Primary 55P62; Secondary 55M30. Key words and phrases: rational homotopy, coformal spaces, 2-cones, differential Ext. \begin{matrix}c \\ ◯ \end{matrix} 2004, J.-B. Gatsinzi. Permission to copy for private use granted.

where (𝕃(W ⊕ sW), d) is acyclic. Moreover the differential on 𝕃(V ⊕ sW) verifies δsW ⊂ 𝕃(V). Hence a 2-cone X has a Quillen model of the form (𝕃(V_{1} ⊕ V_{2}), δ) such that δV_{1} = 0 and δV_{2} ⊂ 𝕃(V_{1}).

A Sullivan model of a space X is a cochain algebra (∧Z, d) that algebraically models the rational homotopy type of X. In particular, one has an isomorphism of graded algebras H^{∗}(∧Z, d) ≅ H^{∗}(X; ℚ). The model is called minimal if dZ ⊂ ∧^{⩾2}Z. In this case the vector spaces Z^{n} and Hom(π_{n}(X), ℚ) are isomorphic. If X has the rational homotopy type of a finite CW-complex, we say that X is elliptic if Z is finite dimensional, otherwise X is called hyperbolic.

2. Models of classifying spaces

Henceforth X will denote a simply connected finite CW-complex and L_{X} its homotopy Lie algebra. Let aut X denote the space of free self homotopy equivalences of X, aut_{1}(X) the path component of aut X containing the identity map of X. The space Baut_{1}(X) classifies fibrations with fibre X over simply connected base spaces [4].

The Schlessinger-Stasheff model for Baut_{1}(X) is defined as follows [12]. If (𝕃(V), δ) is a Quillen model of X, we define a differential Lie algebra Der𝕃(V) = ⊕_{k⩾1} Der_{k} 𝕃(V) where Der_{k} 𝕃(V) is the vector space of derivations of 𝕃(V) which increase the degree by k, with the restriction that Der_{1} 𝕃(V) is the vector space of derivations of degree 1 that commute with the differential δ.

Define the differential Lie algebra (s𝕃(V) ⊕_{∼} Der 𝕃(V), D) as follows:

• The graded vector space s𝕃(V) ⊕_{∼} Der 𝕃(V) is isomorphic to s𝕃(V)⊕Der 𝕃(V),

• If θ, γ ∈ Der 𝕃(V) and sx, sy ∈ s𝕃(V), then [θ, γ] = θγ −(−1)^{|θ||γ|}γθ, [θ, sx] = (−1)^{|θ|}sθ(x) and [sx, sy] = 0,

• The differential D is defined by Dθ = [δ, θ], D(sx) = −sδx + ad x, where ad x is the inner derivation determined by x.

From the Sullivan minimal model (∧Z, d), Sullivan defines the graded differential Lie algebra (Der ∧Z, D) as follows [13]. For k > 1, the vector space (Der ∧Z)_{k} consists of the derivations on ∧Z that decrease the degree by k and (Der ∧Z)_{1} is the vector space of derivations of degree 1 verifying dθ + θd = 0. For θ, γ ∈ Der ∧V, the Lie bracket is defined by [θ, γ] = θγ − (−1)^{|θ||γ|}γθ and the differential D is defined by Dθ = [d, θ].

We have the following result:

Theorem 1. [13, 12, 14] The differential Lie algebras (Der ∧Z, D) and (s𝕃(V) ⊕_{∼} Der 𝕃(V), D) are models of the classifying space B aut_{1}(X).

An indirect proof of the Schlessinger-Stasheff model is given in [8, Theorem 2].

3. The classifying space spectral sequence

Recall that if (L,δ) is a graded differential Lie algebra, then L becomes an UL module by the adjoint representation ad : L → Hom(L, L). In the sequel all Lie algebras are endowed with the above module structure.

Let (𝕃(V), δ) be a Quillen model of a finite CW-complex and (T V, d) its enveloping algebra. On the T V -module T V ⊗ (ℚ ⊕ sV), define a ℚ-linear map

S : T V ⊗ (ℚ ⊕ sV) → T V ⊗ (ℚ ⊕ sV)

as follows:

• S(1 ⊗ x) = 0 for all x ∈ ℚ ⊕ sV,

• S(v ⊗ 1) = 1 ⊗ sv for all v ∈ V,

• If a ∈ T V and x ∈ T V ⊗ (ℚ ⊕ sV) with |x| > 0, then S(a.x) = (−1)^{|a|}a.S(x). The differential on the T V -module T V ⊗ (ℚ ⊕ sV) is defined by

D(1 ⊗ sv) = v ⊗ 1 − S(dv ⊗ 1) for v ∈ V and D(1 ⊗ 1) = 0.

It follows from [1] that (T V ⊗(ℚ⊕sV), D) is acyclic, hence it is a semifree resolution of ℚ as a (T V, d)-module [6, §6].

Using the Schlessinger-Stasheff model of the classifying space, the author proved the following:

Theorem 2. [8] The differential graded vector spaces Hom_{T V} (T V ⊗(ℚ⊕sV), 𝕃(V)) and s𝕃(V) ⊕_{∼} Der 𝕃(V) are isomorphic. Moreover, for n ⩾ 0, the ℚ-vector spaces Ext_{T V}^{n} (ℚ, 𝕃(V)) and π_{n+1}(ΩB aut_{1} X) ⊗ ℚ are isomorphic.

In particular if X is a coformal space, one has an isomorphism π_{n}(B aut_{1} X)⊗ℚ ≅ Ext_{UL_{X}}^{n} (ℚ, L_{X}). Therefore π_{∗}(B aut_{1} X) ⊗ ℚ can be computed by the means of a projective resolution of ℚ as an UL_{X}-module.

Consider the complex P = Hom_{T V} (T V ⊗ (ℚ ⊕ sV), 𝕃(V)). Filter V as follows

F_{0}V = 0, F_{p+1}V = {x ∈ V : dx ∈ 𝕃(F_{p}V)}.

We will denote V_{p} = F_{p}V/F_{p−1}V. If F_{n−1}V ≠ F_{n}V = V, following Lemaire [10] we say that V is of length n. We will restrict to spaces with a Quillen model of length n.

Define a filtration on P = T V ⊗ (ℚ ⊕ sV) as follows:

P_{0} = T V ⊗ ℚ, P_{1} = T V ⊗ (ℚ ⊕ sV_{1}), …, P_{n} = T V ⊗ (ℚ ⊕ sV_{⩽n}).

We filter the complex

Hom_{T V} (T V ⊗ (ℚ ⊕ sV), 𝕃(V))

by

F_{k} = {f : f(P_{k−1}) = 0}.

This yields a spectral sequence E_{r} such that E_{1}^{p,q} = Hom_{ℚ}(sV_{p}, L_{X}) for p > 1, E_{1}^{0,q} = Hom_{ℚ}(ℚ, L_{X}) and that converges to Ext_{T V}^{∗} (ℚ, 𝕃(V)). This sequence will be called the classifying space spectral sequence of X.

Now assume that X is coformal and let A = UL_{X}. If 𝕃(V_{1})/I is a minimal presentation of L_{X}, then there is a quasi-isomorphism (𝕃(V_{1}⊕V_{2}⊕⋯⊕V_{n}), δ) → L_{X} which extends to p : (T V, d) ⟶^{≃} (A, 0). The (E_{1}, d) term provides a resolution

⋯ → A ⊗ sV_{n} → A ⊗ sV_{n−1} → ⋯ → A ⊗ sV_{1} → A → ℚ

of ℚ as an A-module. Here the differential is given by the composition

sV_{n} ⟶^{D} T V ⊗ (ℚ ⊕ sV_{n−1}) ⟶^{p⊗id} A ⊗ (ℚ ⊕ sV_{n−1}).

The spectral sequence will therefore collapse at E_{2} level. Moreover Ext_{A}^{∗}(ℚ, L_{X}) is endowed with a Lie algebra structure verifying

[Ext^{p,∗}, Ext^{q,∗}] ⊂ Ext^{p+q−1,∗} .  (2)

The Lie bracket can be defined using the bijection between the Koszul complex C^{∗}(L_{X}, L_{X}) and derivations on the Sullivan model C^{∗}(L_{X}, ℚ) of X [9, Proposition 4] (see also [7] for a direct definition of the Lie bracket on C^{∗}(L_{X}, L_{X})). Alternatively one may use the bijection

Hom_{T V} (T V ⊗ (ℚ ⊕ sV), 𝕃(V)) ≅ s𝕃(V) ⊕_{∼} Der 𝕃(V)

to transfer a Lie algebra structure on Hom_{T V} (T V ⊗ (ℚ ⊕ sV), 𝕃(V)) from

s𝕃(V) ⊕_{∼} Der 𝕃(V).

Definition 3. Let L be a Lie algebra. An element x ∈ L is called locally nilpotent if for every y ∈ L, there is a positive integer k such that (ad x)^{k}(y) = 0. A subset K ⊂ L is called locally nilpotent if each element of K is locally nilpotent.

We deduce from Equation (2) the following

Proposition 4. Let X be a coformal space of homotopy Lie algebra denoted L_{X}. If X has a Quillen model (𝕃(V), δ), of length n, one has:

1. For k ≠ 1, Ext_{A}^{k}(ℚ, L_{X}) is locally nilpotent,

2. Ext_{A}^{1}(ℚ, L_{X}) is a subalgebra of Ext_{A}(ℚ, L_{X}),

3. If Ext_{A}^{0}(ℚ, L_{X}) = 0, then ⊕_{i⩾i_{0}} Ext_{A}^{i}(ℚ, L_{X}) is an ideal of Ext_{A}(ℚ, L_{X}), for i_{0} ⩾ 1.

We will now assume that X is a coformal 2-cone. Recall that X has a Quillen minimal model of the form (𝕃(V_{1} ⊕V_{2}), δ), with δV_{1} = 0 and δV_{2} ⊂ 𝕃(V_{1}). Moreover π_{∗}(ΩX)⊗ℚ = H_{∗}(𝕃(V_{1} ⊕V_{2}), δ) = 𝕃(V_{1})/I, where I is the ideal of 𝕃(V_{1}) generated by δV_{2}.

Definition 5. Let 𝕃(V) be a free Lie algebra where {a, b, c, … } is a basis of V. Denote 𝕃^{n}(V) the subspace of 𝕃(V) consisting of Lie brackets of length n. Consider a basis {u_{1}, u_{2}, … } of 𝕃^{n}(V) where each u_{i} is a Lie monomial. If x ∈ {a, b, c, … }, we define the length of u_{i} in the variable x, l_{x}(u_{i}), as the number of occurrences of the letter x in u_{i}. If u = ∑ r_{i}u_{i} ∈ 𝕃^{n}(V), define l_{x}(u) = min{l_{x}(u_{i})} and if v = ∑ v_{i} where v_{i} ∈ 𝕃^{i}(V), l_{x}(v) = min{l_{x}(v_{i})}.

It is straightforward that the above definition extends to the enveloping algebra T (V).

Theorem 6. Let X be a coformal 2-cone and (𝕃(V_{1} ⊕V_{2}), δ) be its Quillen minimal model. Choose a basis {x_{1}, x_{2}, … } for V_{1} and a basis {y_{1}, y_{2}, … } for V_{2}. If for some x_{k} ∈ {x_{1}, x_{2}, … }, l_{x_{k}}(δy_{j}) ⩾ 2 for all y_{j} ∈ {y_{1}, y_{2}, … }, then Ext_{A}^{2,∗}(ℚ, L_{X}) is infinite dimensional.

Proof. Note that for i ≠ k the element (ad x_{i})^{n}(x_{k}) is a nonzero homology class in H_{∗}(𝕃(V_{1} ⊕ V_{2}), δ) as it contains only one occurrence of x_{k}. Take y_{t} ∈ {y_{1}, y_{2}, … } and x_{m} ∈ {x_{1}, x_{2}, … } with m ≠ k. For each n ⩾ 1, define f_{n} ∈ Hom_{A}(A ⊗ sV_{2}, L_{X}) by f_{n}(sy_{t}) = (ad x_{m})^{n}(x_{k}) and f_{n}(sy_{j}) = 0 for j ≠ t. Obviously f_{n} ∈ Hom_{A}(A ⊗ sV_{2}, L_{X}) is a cocycle. Suppose that f_{n} is a coboundary. There exists g_{n} ∈ Hom_{A}(A ⊗ sV_{1}, L_{X}) such that f_{n}(sy_{t}) = g_{n}(dsy_{t}). From the definition of the differential d, one has dsy_{t} = ∑_{i} p_{i}sx_{i}, where the p_{i}’s are polynomials in the variables x_{1}, x_{2}, …. From the hypothesis on the differential dy_{t} one knows that l_{x_{k}}(p_{i}) ⩾ 2 for i ≠ k and l_{x_{k}}(p_{k}) ⩾ 1. By using the number of occurrences of the variable x_{k}, one deduces from the previous equalities that (ad x_{m})^{n}(x_{k}) equals the component of length 1 in x_{k} of p_{k}g_{n}(sx_{k}). Therefore, in the monomial decomposition of g_{n}(sx_{k}) (resp. p_{k}) there must exist (ad x_{m})^{n−s}(x_{k}) (resp. x_{m}^{s}). We obtain a contradiction with l_{x_{k}}(p_{k}) ⩾ 1.

The cocycles f_{n} create an infinite number of non zero classes (of distinct degrees) and the space Ext_{A}^{2,∗}(ℚ, L_{X}) is infinite dimensional.

Corollary 7. If hypotheses of the above theorem are satisfied, then cat(B aut_{1}(X)) = ∞.

Proof. If sx ∈ Ext^{0,∗} ⊂ 𝕃(V_{1})/I and f ∈ Ext^{2,∗} then [f, sx] = ±sf(x). As elements of Ext^{2,∗} vanish on V_{1}, we deduce that [Ext^{2,∗}, Ext^{0,∗}] = 0. It follows from Theorem 6 that J = Ext_{UL_{X}}^{2} (ℚ, L_{X}) is an infinite dimensional ideal of π_{∗}(ΩB aut_{1}(X)). Moreover it follows from Equation (2) that J is abelian. We deduce that the category of B aut_{1}(X) is infinite [5, Theorem 12.2].

If X is an elliptic space of Sullivan minimal model (∧Z, d) then Der ∧ Z is a finite dimensional ℚ-vector space. Hence the homotopy Lie algebra of B aut_{1}(X) is finite dimensional, therefore π_{∗}(ΩB aut_{1}(X)) ⊗ ℚ is nilpotent. In [11], P. Salvatore proved that if X = S^{2n+1} ∨ S^{2n+1}, then π_{∗}(ΩB aut_{1}(X)) ⊗ ℚ contains an element α that is not locally nilpotent. The proof consists in the construction of two outer derivations α and β of the free Lie algebra 𝕃(a, b), where |a| = |b| = 2n, such that (ad α)^{i}(β) ≠ 0, for every integer i > 0. The technique can be applied to any free Lie algebra with at least two generators. Therefore π_{∗}(ΩB aut_{1}(X)) ⊗ ℚ contains an element α that is not locally nilpotent if X is a wedge of two spheres or more.

P. Salvatore asked if π_{∗}(ΩB aut_{1}(X)) ⊗ ℚ has always such a property for every hyperbolic space X. A positive answer to this question would provide another characterization of the elliptic-hyperbolic dichotomy [5].

For a product space we have the following

Proposition 8. If X = Y × Z is a product space such that the Lie algebra π_{∗}(ΩB aut_{1}(Y)) ⊗ ℚ is not nilpotent, then π_{∗}(ΩB aut_{1}(X)) ⊗ ℚ is not nilpotent.

Proof. Let (∧V, d) and (∧W, d′) be Sullivan models of Y and Z respectively. Therefore (∧V ⊗ ∧W, d ⊗ d′) is a Sullivan model of X. It follows from [12] that

H_{∗}(Der(∧V ⊗ ∧W))≅H_{∗}(Der ∧ V) ⊗ H^{∗}(∧W) ⊕ H^{∗}(∧V) ⊗ H_{∗}(Der ∧ W).

Therefore π_{∗}(ΩB aut_{1}(Y)) ⊗ ℚ is a subalgebra of π_{∗}(ΩB aut_{1}(X)) ⊗ ℚ.

In particular if Y is a wedge of at least two spheres, then the Lie algebra π_{∗}(ΩB aut_{1}(Y)) ⊗ ℚ is not nilpotent and so is π_{∗}(ΩB aut_{1}(X)) ⊗ ℚ.

We can extend Salvatore’s result to some certain types of coformal hyperbolic 2-cones.

Theorem 9. Under the hypotheses of Theorem 6, the rational homotopy Lie algebra of B aut_{1}(X) is not nilpotent.

Proof. For i ≠ k, let (ad x_{k})^{n}(x_{i}) be a nonzero element of L_{X}. Define α_{n} ∈ Ext_{A}^{1}(ℚ, L_{X}) by α_{n}(sx_{i}) = (ad x_{k})^{n}(x_{i}) and zero on the other generators of L_{X}. Take w ∈ V_{2} and define β_{m} ∈ Ext_{A}^{2}(ℚ, L_{X}) by β_{m}(sw) = (ad x_{k})^{m}(x_{i}) and zero elsewhere. A short computation shows that [α_{n}, β_{m}] = ±β_{m+n}. Hence (ad α_{n})^{l}(β_{m}) ≠ 0 for all l ⩾ 1. Therefore π_{∗}(ΩB aut_{1}(X)) ⊗ ℚ is not nilpotent.

Example 10. Consider the space X for which the Quillen minimal model is (𝕃(a, b, c), d) with da = db = 0 and dc = [b, [b, a]]. The space X satisfies the hypothesis of Theorem 6. Therefore cat(B aut_{1}(X)) is infinite. Moreover the homotopy Lie algebra of B aut_{1}(X) ⊗ ℚ is not nilpotent.

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J.-B. Gatsinzi gatsinzj@mopipi.ub.bw

Department of Mathematics, University of Botswana, Private Bag 0022, Gaborone, Botswana