RATIONAL CATEGORY AND CONE LENGTH OF POINCARÉ COMPLEXES

OCTAVIAN CORNEA, YVES FÉLIX, AND JEAN-MICHEL LEMAIRE

Abstract. We prove the so-called Lemaire-Sigrist conjecture for 1-connected Poincaré complexes, namely that for such spaces rational Lusternik-Schnirelmann category equals rational cone length. We also prove that the attachement of the top cell in such a space raises rational L.-S. category by one.

1. Introduction

The Lusternik-Schnirelmann category of a topological space X, cat(X), is the least integer n such that X can be covered by n + 1 open sets contractible in X. This a well-known and much studied homotopy invariant (see [10], [11]). A closely related invariant, called strong category, Cat(X), is the least integer n such that there exists a space X′ in the homotopy type of X which can be covered by n + 1 open sets contractible in themselves. This invariant was considered by Ganéa and Takens [17] who also proved that for any connected space X one has

cat(X) ≤ Cat(X) ≤ cat(X) + 1 .

For many years, the only known examples for which Cat(X) = cat(X)+ 1 were co-H-spaces (i.e. spaces of category 1) which are not suspensions (i.e. spaces of cone length 1). But since a simply connected rational co-H-space has the rational homotopy type of a wedge of rational spheres, it was doubtful that such examples would exist in rational homotopy.

For a connected space X, strong category is now known [2] to coincide with the cone length cl(X), namely the least integer n such that there is a sequence of cofibrations

Σ^{i}A_{i} ⟶X_{i−1} ⟶X_{i} , i = 1, ...n ,

2000 Mathematics Subject Classification. 55M30, 55P62, 57P10.

Key words and phrases. Lusternik-Schnirelmann category, cone length, rational homotopy, Poincaré duality.

This research is partially supported by a NATO grant. The third author is partially supported by CNRS.

where X_{0} is a point, X_{n} has the homotopy type of X and Σ^{i}A_{i} is the i-fold suspension of some space A_{i}.

Let us recall that the rational category cat_{0}(X) of a 1-connected cwspace X is defined as the Lusternik-Schnirelmann category cat(X_{0}) of its 0-localization. One may define its rational cone-length cl_{0}(X) in the same way. It was conjectured in [13] that cat_{0}(X) = cl_{0}(X) for any 1-connected cw-space X.

A remarkable counterexample to this conjecture has recently been obtained by N. Dupont [3] : he has produced a finite cw-complex Z with cl_{0}(Z) = 4, but such that cl_{0}(Z ∨ S^{N}) = 3 for some sphere S^{N}, so that cat_{0}(Z) = 3.

In this paper, we prove that such a phenomenon cannot occur for closed manifolds. More precisely, we prove:

Theorem 1.1. Let X be a simply-connected space whose rational cohomology has finite type and satisfies Poincaré duality, then cat_{0}(X) = cl_{0}(X)

Indeed, by a recent result of Steve Halperin and two of us [5], the rational category of simply connected Poincaré duality spaces is also equal to the “Toomer invariant” e_{0}(X). Here we shall give a direct proof of this result, which yields another interesting property of Poincaré complexes :

Theorem 1.2. Let X be a simply-connected N-dimensional CW complex whose rational cohomology has finite type and satisfies Poincaré duality. Let Y be the (N − 1)-skeleton of X, so that X has the rational homotopy type of Y ∪ e^{N}. Then cat_{0}(X) = cat_{0}(Y) + 1.

Note that J.J. Rivadeneyra Perez [15] has proven that if X is a PL compact closed manifold and p is a point in X , then cat(X\p) ≤ cat(X) , at least when cat(X) ≥ 2. In other words, attaching the top cell in a PL manifold does not decrease category. Our result says that the rational category does increase by 1.

2. Rational L.-S. category and Toomer’s invariant

Let X be a 1-connected cw-space. The rational category of X, cat_{0}(X), i.e. the category of the rationalization X_{0} of X, was first considered by G. Toomer, who proved the inequality cat_{0}(X) ≤ cat(X) [19]. Indeed this inequality readily follows by localizing Ganéa’s fibrecofibre construction, and the same argument shows that cl_{0}(X) ≤ cl(X).

A weaker lower bound for category, noted e_{0}(X), was also introduced by Toomer in [18]. Let B(A) denote the reduced bar construction on a (connected) chain algebra A. Since the chain coalgebras B(C_{∗}(ΩX; ℚ)) and C_{∗}(X; ℚ) are quasi-isomorphic, the filtration of B(C_{∗}(ΩX; ℚ)) by tensor powers yields a spectral sequence, which converges to H_{∗}(X; ℚ). We then define :

e_{0}(X) = max{p |E_{p,∗}^{∞} ≠ O}.

In [18], Toomer proves the inequality e_{0}(X) ≤ cat_{0}(X).

As it is well-known ([14], [16], [8]), there is an equivalence of homotopy categories between the category of simply connected rational spaces with finite type rational cohomology, and the category of 1connected commutative differential graded algebras of finite type over ℚ. To every space X corresponds a commutative differential graded algebra of the form (ΛV, d), with d(V) ⊂ Λ^{≥2}(V), unique up to isomorphism. This algebra is called the Sullivan minimal model of X, and it uniquely determines the rational homotopy type of X. In particular cat_{0}(X) and e_{0}(X) can be defined in terms of the Sullivan minimal model of X.

Let us denote by p_{n} : (ΛV, d) → (ΛV/Λ^{>n}V, d̄) the canonical projection, and by

f_{n} : (ΛV ⊗ (ℚ ⊕ E), d) ⟶^{≃} (ΛV/Λ^{>n}V, d̄)

a semifree resolution of (ΛV/Λ^{>n}V, d̄) as a (ΛV, d) differential module (cf. [6] for the definition and properties of semifree resolutions). We can moreover assume that f_{n} extends p_{n} through the canonical injection j_{n} : (ΛV, d) → (ΛV ⊗ (ℚ ⊕ E), d), defined by j_{n}(a) = a ⊗ 1, so that we have the commutative diagram :

(ΛV, d)  ⟶^{p_{n}} (ΛV/Λ^{>n}V, d̄)
j_{n} ↓  ↗ f_{n}
(ΛV ⊗ (ℚ ⊕ E), d)

Then e_{0}(X) is the least integer n such that H^{∗}(p_{n}) is injective, and cat_{0}(X) is the least integer n′ such that j_{n′} admits a retraction ρ_{n′} in the category of (ΛV, d) differential modules ([4], [9]). The inequality e_{0}(X) ≤ cat_{0}(X) is obvious from these definitions.

3. Poincaré duality spaces

A graded connected finite dimensional algebra A defined over ℚ satisfies Poincaré duality if there is an element ω ∈ A^{N} such that A^{N} = ℚ.ω, A^{>N} = 0 and that the multiplication induces a non-degenerated bilinear form

A ⊗ A ⟶ A ⟶^{ε} ℚ; ε(A^{<N}) = 0 , ε(ω) = 1 .

In other words the map (of degree −N) ν : A → A^{∨} = Hom_{ℚ}(A, ℚ) defined by ν(a)(b) = (−1)^{N|a|}ε(a · b) is an isomorphism.

In all that follows X will denote a simply connected N-dimensional finite CW complex whose rational cohomology satisfies Poincaré duality. Up to rational equivalence we may assume that X = Y ∪_{φ} e^{N}, with dim Y ≤ N − 1. Let n = e_{0}(X), Ω ∈ H^{N}(X; ℚ) be a fundamental class, and (ΛV, d) be the Sullivan minimal model of X. We then have :

Lemma 3.1. [4] e_{0}(X) is the least integer n such that Ω can be represented by a cocycle ω in Λ^{≥n}V. □

We consider the subspace E ⊂ (ΛV)^{N} defined by

E = (Im d)^{N} + (Λ^{>n}V)^{N} .

By definition of the integer n, the element ω does not belong to E. We denote by S a linear complement of E ⊕ ℚω in (ΛV)^{N},

(ΛV)^{N} = E ⊕ ℚω ⊕ S .

The linear map f : ΛV → ℚ of degree −N defined by f(ω) = 1, f(E ⊕ S) = 0, and f(ΛV)^{i} = 0 for i ≠ N is clearly a cocycle in the cochain complex

((ΛV)^{∨}, ∂) = (Hom(ΛV, ℚ), ∂) , with ∂(g)(x) = −(−1)^{|g|}g(dx) .

We now define a linear map of degree −N

ν : ΛV → (ΛV)^{∨}

by putting

ν(α)(β) = (−1)^{N|α]}f(α · β) .

We observe that (ΛV)^{∨} admits a canonical structure of (ΛV, d)- module, defined by

∀γ ∈ ΛV ∀φ ∈ (ΛV)^{∨} , γ · φ(β) = (−1)^{|γ| |φ|}φ(γ · β).

We then have :

Lemma 3.2. The map ν is a degree −N quasi-isomorphism of differential (ΛV, d)-modules.

Proof. The commutation of ν with the differential and the (ΛV, d)- structure are easy calculations. The map ν is a quasi-isomorphism because it induces the Poincaré duality isomorphism in cohomology

ν : H^{∗}(X; ℚ) ⟶^{≅} Hom(H^{∗}(X; ℚ), ℚ). □

We can now prove Theorem 1.2.

Proof. Let (ΛW, d) be the minimal model of the space Y and j : (ΛV, d) → (ΛW, d) be the minimal model of the inclusion i : Y → X. If R denotes a linear complement of the cocycles in (ΛV)^{N−1}, we can write the following commutative diagram, in which the horizontal arrows are quasi-isomorphisms :

(ΛV, d) ⟶^{≃} (ΛV/((ΛV)^{>N} ⊕ E ⊕ S ⊕ R), d̄)
j↓  ↓
(ΛW, d) ⟶^{ρ}  (ΛV/((ΛV)^{≥N} ⊕ R), d̄)

Remark that ν(ω) is the augmentation map ε : ΛV → ℚ and that ℚε is a (ΛV, d) -submodule of ((ΛV)^{∨}, ∂). Note that ν(Λ^{>n}V) = 0 = ν(E ⊕ S ⊕ R)) because f has been defined accordingly. Therefore ν induces a quotient quasi-isomorphism

ν̄ : (ΛV/((ΛV)^{≥N} ⊕ R), d̄) ⟶ ((ΛV)^{∨}/ℚε, ∂)

and the map ν̄ρ factors through the quotient algebra (ΛW/Λ^{≥n}W, d̄) as a map ψ :

(ΛW, d)  ⟶^{ν̄ρ} ((ΛV)^{∨}/ℚε, ∂)
↓p  ↗ψ
(ΛW/Λ^{≥n}W, d̄)

Denote by f_{n} : (ΛW ⊗ (ℚ ⊕ E), D) ⟶^{≃} (ΛW/Λ^{≥n}W, d̄) a semifree resolution of (ΛW/Λ^{≥n}W, d̄) as a (ΛW, d)-module, which extends the projection p :

(ΛW, d)  ⟶^{p} (ΛW/Λ^{≥n}W, d̄)
q ↓  ↗ f_{n}
(ΛW ⊗ (ℚ ⊕ E), D)

Since ν̄ρ is a quasi-isomorphism, there is a map of (ΛW, d)-modules

μ : (ΛW ⊗ (ℚ ⊕ E), D) → (ΛW, d)

such that μq = id_{ΛW} and ν̄ρμ ∼ ψf_{n} :

(ΛW, d)  ⟶^{ν̄ρ} ((ΛV)^{∨}/ℚε, ∂)
q ↓↑ μ  ψf_{n} ↗
(ΛW ⊗ (ℚ ⊕ E), D)

The existence of μ means that cat_{0}(Y) ≤ n − 1. Thus n = e_{0}(X) ≥ cat_{0}(Y) + 1. But since X = Y ∪ e^{N} we have cat_{0}(X) ≤ cat_{0}(Y) + 1 ≤ e_{0}(X). Since cat_{0}(X) ≥ e_{0}(X) for all X, we get cat_{0}(X) = cat_{0}(Y) + 1 = e_{0}(X). Both Theorem 1.2 and the equality cat_{0}(X) = e_{0}(X) are proven. □

4. The rational cone length of a Poincaré duality space

In [1] one finds the following result :

Theorem 4.1. Let X be a simply connected space of rational LS category n, then there exists a wedge of spheres Σ such that the rational cone length of X ∨ Σ is equal to n.

From this theorem we deduce the following proposition :

Proposition 4.2. Let X be a space of rational LS category n and let Z be obtained by attaching cells to X along a map φ : ⋁_{i} S^{n_{i}} → X. If cat_{0}(Z) = n + 1 then cl_{0}(Z) = n + 1.

Proof. By 4.1 there is a wedge of spheres Σ such that cl_{0}(X ∨ Σ) = n. By definition Z is the mapping cone of φ, but it is also the mapping cone of the map

φ ∨ id_{Σ} : ⋁_{i} S^{n_{i}} ∨ Σ ⟶X ∨ Σ .

Therefore cl_{0}(Z) ≤ n + 1 = cat_{0}(Z), hence the equality since cl_{0}(Z) ≥ cat_{0}(Z). □

Theorem 1.1 now is a direct consequence of Theorem 1.2 and Proposition 4.2.

References

[1] Cornea, Octavian, Cone length and Lusternik-Schnirelmann category, Topology 33 (1994), 95-111.

[2] Cornea, Octavian, Strong LS category equals cone-length, Topology 34, No.2, 377-381 (1995).

[3] Dupont, Nicolas, A counterexample to the Lemaire-Sigrist conjecture, preprint (1996).

[4] Félix, Yves, Halperin, Stephen, Rational L.-S. category and its applications, Trans. Amer. Math. Soc. 273 (1982), 1-37.

[5] Félix, Yves, Halperin, Stephen and Lemaire, Jean-Michel, The rational LS category of products and Poincaré duality complexes, preprint (1996).

[6] Félix, Yves, Halperin, Stephen and Thomas, Jean-Claude, Differential graded algebras in topology, in Handbook of Algebraic Topology, North-Holland (1995) 829-865.

[7] Halperin, Stephen, Finiteness in the minimal models of Sullivan, Trans. Amer. Math. Soc. 230 (1977), 173-199.

[8] Halperin, Stephen, Lectures on Minimal Models, Mém. Soc. Math. France 9/10 (1983).

[9] Hess, Kathryn, A proof of Ganea’s conjecture for rational spaces, Topology 30 (1991), 205-214.

[10] James, Ioan, On category in the sense of Lusternik-Schnirelmann , Topology 17 (1978), 331-348.

[11] James, Ioan, Lusternik-Schnirelmann category, in Handbook of Algebraic Topology, North-Holland (1995), 1293-1310.

[12] Jessup, Barry, L.-S. category and homogeneous spaces, J. Pure Appl. Algebra 65 (1990), 45-56.

[13] Lemaire, Jean-Michel and Sigrist, François, Sur les invariants d’homotopie rationnelle liés à la L.-S. catégorie, Comm. Math. Helv. 56 (1981), 103-122.

[14] Quillen, Daniel, Rational homotopy theory, Ann. Math.90 (1969), 205-295.

[15] Rivedeneyra Perez, J.J., On cat(X\p), Intern. J. Math. and Math. Sci. 15 (1992), 465-468.

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

[17] Takens, Floris, The Lusternik-Schnirelmann categories of a product space, Compos. Math. 22 (1970), 175-180.

[18] Toomer, Graham H., Lusternik-Schnirelmann category and the Moore spectral sequence, Math. Z. 138 (1974), 123-143.

[19] Toomer, Graham H., Topological localization, category and cocategory, Canadian J. Math. 27 (1975), 323-329.

Géométrie, Algèbre et Topologie, URA CNRS 751, Université des Sciences et Technologies de Lille, 59655 Villeneuve d’Ascq Cedex, France

E-mail address: cornea@gat.univ-lille1.fr

Institut Mathématique, 2, Chemin du cyclotron, 1348 Louvain-La-Neuve, Belgique

E-mail address: felix@agel.ucl.ac.be

Laboratoire de Mathématiques Jean Dieudonné, Unité Mixte de Recherches CNRS – Université de Nice Sophia-Antipolis (UMR 6621), Parc Valrose, F-06108 Nice Cedex 2, France

Current address: Directeur adjoint, Département Sciences Physiques et Mathématiques, CNRS, 3 rue Michel-Ange, F-75794 Paris Cedex 16, France

E-mail address: lemaire@math.unice.fr