Lusternik-Schnirelmann category of classifying spaces.

J.-B. Gatsinzi ^{∗}

Abstract Let X be a finite simply connected CW-complex. In this paper, we show that the Lusternik-Schnirelmann category of the classifying space B aut X is infinite if X = S^{n} ∨ Y.

Mathematical Subject Classification: 55 P 62, 55 M 30.

Key words and phrases: Lusternik-Schnirelmann category, rational homotopy, classifying space.

1 Introduction

In this paper X will denote a simply connected CW-complex of finite type, that is, H^{n}(X, ℚ) is a finite dimensional ℚ-vector space, for each n. Recall that the Lusternik-Schnirelmann category of a topological space, cat(X), is the least integer n such that X can be covered by (n + 1) open subsets contractible in X, and is infinite if no such n exists.

Denote by X_{0} the localization of X at zero, the rational Lusternik-Schnirelmann category, cat_{0}(X), is defined by cat_{0}(X) = cat(X_{0}). It verifies cat_{0}(X) ≤ cat(X) [5].

In this paper we will use the theory of Quillen models. The Quillen minimal model of X is a free chain Lie algebra (𝕃(V), δ) satisfying δV ⊂ 𝕃^{≥2}V and the graded vector space V is related to the cohomology of X by V_{n} ≅ H^{n+1}(X, ℚ) [9, 1].

∗Supported by the Max-Planck Institut für Mathematik, Bonn

Fibrations of fiber in the homotopy type of X are obtained, up to fiber homotopy equivalence, as pull back of the universal fibration X ⟶ B aut^{•} X ⟶ B aut X [3, 4]; here aut X denotes the topological monoid of all self-homotopy equivalences of X, aut^{•}X is the submonoid of aut X consisting of pointed self-homotopy equivalences of X, and B is the Dold-Lashof functor [2]. Let B̃ aut X ⟶^{f} B aut X be the universal covering, the induced fibration X ⟶ B̃ aut^{•}X ⟶ B̃ aut X is universal for fibrations with simply connected base spaces [4, Proposition 4.2].

This work deals with the calculation of the Lusternik-Schnirelmann category of B aut X under restrictions on X. The computation of cat(B aut X) is of great interest as shown by the next result.

Proposition 1 Let X be a 1-connected CW-complex and G a connected compact Lie group acting on X. If the Lusternik-Schnirelmann category of B̃ aut X is finite, then the Borel fibration X ⟶ EG ×_{G} X ⟶ BG is rationally trivial.

Proof. Let f : BG ⟶ B̃ aut X be the classifying map of the Borel fibration X ⟶ EG×_{G}X ⟶ BG. Consider the map H^{∗}(f, ℚ) : H^{∗}(B̃ aut X, ℚ) ⟶ H^{∗}(BG, ℚ) = ΛV, where V is concentrated in even degrees. Suppose now that cat(B̃ aut X) is finite. Then H^{∗}(f, ℚ) is trivial, otherwise the nilpotency index of H^{∗}(B̃ aut X, ℚ) is infinite. It is then easily seen that the map f : BG ⟶ B̃ aut X is rationally trivial.

We briefly recall the construction of a model of B̃ aut X that we will use to compute cat(B̃ aut X).

Define the Lie algebra of derivations (Der𝕃(V), D) as follows: Der𝕃(V) = ⊕_{k≥1} Der_{k}(𝕃(V)), where Der_{k}(𝕃(V)) is the vector space of derivations which

increase the degree by k with the restriction that Der_{1}(𝕃(V)) is the vector space of derivations of degree one which commute with the differential δ.

Given two derivations θ and θ′, the Lie bracket is defined by [θ, θ′] = θθ′ − (−1)^{|θ||θ′|}θ′θ and the differential D is defined by Dθ = [δ, θ].

Define the differential Lie algebra (s𝕃(V) ⊕_{∼} Der𝕃(V), D̃) as follows:

• s𝕃(V) ⊕_{∼} Der𝕃(V) is isomorphic to s𝕃(V) ⊕ Der𝕃(V) as a graded vector space, • If θ, θ′ ∈ Der 𝕃(V) and sx, sy ∈ s𝕃(V), then [θ, θ′] = θθ′ −(−1)^{|θ||θ′|}θ′θ,

[θ, sx] = (−1)^{|θ|} sθ(x), [sx, sy] = 0,

• D̃(θ) = [δ, θ], D̃(sx) = −sδx + ad x, where ad x is the derivation of 𝕃(V) defined by (ad x)(y) = [x, y].

Theorem 2 [10, 12] A model of the universal fibration X ⟶ B̃ aut^{•}X ⟶ B̃ aut X is given by

(𝕃(V), δ) ⟶ (Der𝕃(V), D) ⟶ (s𝕃(V) ⊕_{∼} Der𝕃(V), D̃).

A model of B̃ aut X from derivations of the Sullivan minimal model of X is described in [11].

We will suppose henceforth that X is a finite simply connected CWcomplex. We know that the LS category of B aut X is not finite in various cases, among them when X is an elliptic space (i.e. π_{∗}(X) ⊗ ℚ is finite dimensional), a wedge of spheres or a product space X = Y × Z [6, 7].

One may expect, by dualty, the LS-category of B aut X be finite when X = Y ∨ Z. We show that it is the case if Z is a wedge of spheres.

2 The theorem

Theorem 3 The Lusternik-Schnirelmann categoy of B aut X is infinite if X = Y ∨ Z, where Z is a wedge of spheres.

Proof of the theorem

Case 1: X = Y ∨ S^{2n}.

Let F ⟶ E ⟶ B be a fibration, then cat(E) ≤ (cat(B)+1).(cat(F)+1)−1. Applying this to the universal fibration X ⟶ B aut^{•} X ⟶ B aut X, we get cat(B aut^{•}X) ≤ (cat(B aut X) + 1).(cat(X) + 1) − 1.

As cat(X) is finite, we deduce that cat(B aut X) is infinite whenever cat(B aut^{•}X) is infinite.

The Quillen minimal model of X is (𝕃(V), δ) = (𝕃(W ⊕ ℚ.x_{2n−1}), δ) where δ(x_{2n−1}) = 0 and δ(W) ⊂ 𝕃(W).

Let θ be the derivation defined by θ(x_{2n−1}) = [x_{2n−1}, x_{2n−1}], θ(W) = 0. Let us show that θ is a cycle in (Der𝕃(V), D). Obviously [δ, θ](x_{2n−1}) = 0 and if w ∈ W, then [δ, θ](w) = δθ(w) + θ(δw) = θ(δw). But δ(w) ∈ 𝕃(W), therefore θ(δw) = 0. Moreover, θ cannot be a boundary. If it is, then there exists a derivation θ′ such that [δ, θ′](x_{2n−1}) = δθ′(x_{2n−1}) = θ(x_{2n−1}) = [x_{2n−1}, x_{2n−1}]; what should imply that [x_{2n−1}, x_{2n−1}] is a boundary in (𝕃(V), δ).

As [θ, θ] = 0, the injection of the Lie subalgebra generated by θ provides a morphism K(ℚ, 2n) ⟶ (B̃ aut^{•} X)_{0} that induces an injective map in homotopy. Therefore, applying the mapping theorem [5], cat(B̃ aut^{•} X) is infinite.

Case 2: X = Y ∨ S^{2n+1}.

Consider the univesal principal G-bundle S^{1} ⟶ P K(ℤ, 2) ⟶ K(ℤ, 2). The circle S^{1} = {z ∈ ℂ, |z| = 1} acts on S^{2n+1} = {(z_{1}, ⋯_{n+1}) ∈ ℂ^{n+1}, |z_{1}|^{2} + ⋯ + |z_{n+1}|^{2} = 1} by z.(z_{1}, ⋯ z_{n+1}) = (zz_{1}, ⋯ zz_{n+1}).

The resulting Borel fibration is not trivial and rationally equivalent to the homotopy fibration S_{0}^{2n+1} ⟶ (ℂP (n))_{0} ⟶ K(ℚ, 2). Extend the action of S^{1} on Y ∨ S^{2n+1} by setting it being trivial on Y. Therefore the resulting Borel fibration is not rationally trivial. Hence, applying Proposition 1, we deduce that cat(B̃ aut X) is infinite.

References

[1] H.J. Baues and J.M. Lemaire, Minimal models in homotopy theory, Math. Ann. 225 (1977), 219-242.

[2] A. Dold and R. Lashoff, Principal quasi-fibrations and fibre homotopy equivalence of bundles, Illinois J. Math. 3 (1959), 285-305.

[3] A. Dold, Halbexakte Homotopiefunktoren, Lecture notes in mathematics, vol. 12 (1966), Springer-Verlag.

[4] E. Dror and A. Zabrodsky, Unipotency and nilpotency in homotopy equivalences, Topology 18 (1979), 187-197.

[5] Y. Félix and S. Halperin, Rational LS-category and its applications, Trans. A.M.S. 273 (1982), 1-17.

[6] J.-B. Gatsinzi, LS category of classifying spaces II, Bull. Belg. Math. Soc. 3 (1996), 243-248.

[7] J.-B. Gatsinzi, The homotopy Lie algebra of classifying spaces, J. Pure and Applied Algebra 120 (1997), 281-289.

[8] J.-B. Gatsinzi, Products and Lusternik-Schnirelmann category of classifying spaces, to appear in Indag. Math.

[9] D. Quillen, Rational homotopy theory, Annals of Math. (2) 90 (1969), 205-295.

[10] M. Schlessinger and J. Stasheff, Deformations theory and rational homotopy type, preprint, 1982.

[11] D. Sullivan, Infinitesimal computations in topology, Publ. I.H.E.S. 47 (1977), 269-331.

[12] D. Tanré, Homotopie rationnelle; modèles de Chen, Quillen, Sullivan, Lecture notes in mathematics, vol. 1025, Springer-Verlag, 1983.

Department of Mathematics University of Botswana Private Bag 0022 Gaborone Botswana e-mail: gatsinzj@noka.ub.bw