Fibrewise Stable Rational Homotopy

Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000

Yves F´elix, Aniceto Murillo and Daniel Tanr´e

Abstract In this paper, for a given space B we establish a correspondence between differential graded ∗ modules over C (B; Q) and fibrewise rational stable spaces over B. This correspondence opens the door for topological translations of algebraic constructions made with modules over a commutative differential graded algebra. More precisely, given fibrations E → B and ′ E → B, the set of stable rational homotopy classes of maps over B is isomorphic to ∗ ∗ ′ ∗ ExtC∗(B;Q)`C (E ; Q),C (E; Q)´. In particular, a nilpotent, finite type CW-complex X is a q rational Poincarécomplex if there exist non trivial stable maps over XQ from (X × S )Q to q+N (X ∨ S )Q for exactly one N.

1. Introduction Sullivan’s approach to rational homotopy theory is based on an adjoint pair of functors between the category of simplicial sets and the category CDGA of commutative differential graded algebras over Q (henceforth called cdga’s ). This correspondence induces an equivalence between the homotopy category of rational (nilpotent and with finite Betti numbers) simplicial sets and a homotopy category of cdga’s. The fact that any algebraic construction or property in CDGA has a topological translation is the most important feature of rational homotopy theory. Until now, however, there was no known procedure for realizing differential graded modules (henceforth called dgm’s) over cdga’s topologically, even though a number of important topological invariants have been interpreted in terms of dgm’s over the years. For instance, by [4], the rational cohomology of a nilpotent space X satisfies Poincaréduality if and only ∗ if the module invariant ExtC∗(X;Q)(Q, C (X; Q)) has dimension one. This fact is one of the motivations of the present paper, in which we establish a general method for a topological realization of the category DGM of dgm’s. We consider maps over a fixed space B. The map is called pointed if it has a fixed section. For each map p: E → B, we denote by p: E+ → B the associated pointed map with E+ = E ∐ B. q The q-th fibrewise suspension of the map p: E → B is the map ΣBE → B, with q q q−1 ΣBE = (E × D ) ∪(E×Sq−1 ) (B × S ). The augmented q-th fibrewise suspension of p is the q-th fibrewise suspension of the map E+ → B, therefore q q q ΣBE+ = (E × D ) ∪(E×Sq−1 ) (B × D ).

If p is a fibration of fibre F , then ΣBE → B is a fibration of fibre ΣF and ΣBE+ → B a fibration of fibre ΣF+, cf. [18, Lemma 6]. Given maps p: E → B and p′ : E′ → B, we use the following notation: ′ ′ • [E, E ]B is the set of homotopy classes of maps over B, i.e., maps f : E → E satisfying p′ ◦ f = p, with homotopy over B,

2000 Mathematics Subject Classiﬁcation 55P62 (primary), 18G15, 55P42, 55R70, 55M30 (secondary). The three authors are partially supported by the MICINN grant MTM2010-18089 Page 2 of 16 YVES FELIX,´ ANICETO MURILLO AND DANIEL TANRE´

′ q q ′ B ′ • [Σ E+, Σ E ] , for q, q > 0, is the set of pointed homotopy classes of maps over B B + B ′ q q ′ ′ ′ ′ B, i.e., maps f : ΣBE+ → ΣB E+ satisfying p ◦ f = p and f ◦ σ = σ , where σ, σ are the canonical sections of the augmented fibrewise suspensions, with homotopy over and under B. By taking iterated augmented fibrewise suspension, we get a sequence of sets of homotopy classes ′ Σ ′ B Σ 2 2 ′ B [E, E ]B → [ΣBE+, ΣBE+]B → [ΣBE+, ΣBE+]B →··· The set of stable fibrewise homotopy classes of maps, of degree r, from E to E′ is the limit

′ r q q+r B E, E = lim [Σ E+, Σ E+]B . B q→+∞ B B In particular, two fibrations, p: E → B and p′ : E′ → B, are called stably rationally equivalent, 0 if they are connected by an isomorphism in E, E′ . B Stable fibrewise homotopy is the subject of the book of Crabb and James [3]. In [3, Proposition 15.8], with some restrictions on the involved spaces, the authors prove that the rational fibrewise stable theory over B can be identified as a fibrewise rational cohomology. In this text, we describe it in terms of an Ext-group in the homotopy category of modules over a cdga model of B. Our main result can be stated as follows.

Theorem 1.1. Let p: E → B and p′ : E′ → B be two fibrations of fibres F and F ′ respectively. We suppose that E, E′ and B are nilpotent spaces with finite Betti numbers ′ and that the fundamental group of B acts locally nilpotently in each Hi(F ; Q) and Hi(F ; Q). Then, we have a bijection

′ r r ∗ ′ ∗ EQ, E =∼ Ext ∗ C (E ; Q), C (E; Q) , Q B C (B;Q) ∗ where C (−; Q) denotes the usual cochain algebra with rational coeﬃcients and −Q the rationalization functor.

The definition of a locally nilpotent action is given in [9], see also [5, Page 197]. In particular, the previous hypothesis are fulfilled if all spaces are simply connected with finite Betti numbers. As a consequence of Theorem 1.1, the two fibrations, p and p′, are stably rationally equivalent if, and only if, there is an isomorphism of C∗(B; Q)-modules between C∗(E; Q) and C∗(E′; Q). This can be stated as follows.

Corollary 1.2. Two ﬁbrations p: E → B and p′ : E′ → B are stably rationally equivalent, if, and only if, C∗(E; Q) and C∗(E′; Q) have isomorphic semifree models as modules over C∗(B; Q).

Lusternik-Schnirelmann category (LS category henceforth) is another invariant which has been characterized by B. Jessup [11] and K. Hess [10] in the homotopy category of DGM. Later on, it was topologically reinterpreted in terms of fibrewise symmetric products [16]. Here, in section 3, we give a new description of stable LS category in terms of fibrewise suspensions. There are two particular fibrations to which we apply Theorem 1.1: the trivial fibration ∗ → X → X, whose augmented q-th fibrewise suspension is the trivial fibration Sq → X × Sq → X, and the path fibration ΩX → PX → X, whose augmented q-th fibrewise suspension is the fibration ΩX ⋉ Sq−1 → X ∨ Sq → X. FIBREWISE STABLE RATIONAL HOMOTOPY Page 3 of 16

Theorem 1.3. Let X be a nilpotent space with ﬁnite Betti numbers. Then there is an isomorphism, r ∗ ∼ r ∼ q q+r XQ ExtC∗(X;Q) Q, C (X; Q) = XQ,PXQ = lim [XQ × S ,XQ ∨ S ]X XQ q→+∞ Q which ﬁts in the commutative diagram ∼ r = / r ∗ XQ,PXQ / Ext ∗ Q Q, C (X; Q) XQ C (X; ) OO ll OO lll OOO lll θ OOO lll ev O' ulll Hr(X; Q). Here, ev is the usual evaluation map [4, 6, 14] and θ associates to a map f : X × Sq → q+r ∗ q+r q X ∨ S the slant product of f [S ] with the generator of Hq(S ).

In particular, Theorem 1.1 provides a new perspective on Poincar´eduality.

Theorem 1.4. Let X be a nilpotent CW-complex of finite LS category and with finite Betti numbers. Then, X is a rational Poincaréduality complex if, and only if, there is an integer N such that: (i) for r 6= N, all pointed stable maps over X, X × Sq → X ∨ Sq+r, are rationally homotopically trivial, (ii) there is, up to a multiple, only one non trivial pointed stable rational map over X, X × Sq → X ∨ Sq+N . In that case, N is the dimension of X.

Theorem 1.5. Let X be a nilpotent CW-complex, with ﬁnite Betti numbers and for which πq(X) ⊗ Q =0 for q large enough. Then, there is an integer N such that: (i) for r 6= N, all pointed stable maps over X, X × Sq → X ∨ Sq+r, are rationally homotopically trivial, (ii) there is, up to a multiple, only one non trivial pointed stable rational map over X, X × Sq → X ∨ Sq+N .

Finally, we point out that ﬁbrewise suspensions appear also in the study of embeddings. 1 2 Let M and X be compact n-dimensional manifolds and let j : M × [ 3 , 3 ] ֒→ X × [0, 1] be an embedding with complement W . Then the projection W ֒→ X × [0, 1] → X makes W a pointed 1 space over X. The composite ν+ : M ∐ X ֒→ (M ×{ 3 }) ∐ (X ×{0}) ֒→ W is a map of pointed X 12 spaces over X. So [ν+] ∈ [M+, W ]X . In [ ], J. Klein shows that [ν+] = 0 if, and only if, there is an embedding of M in X which induces the given embedding j. Our program is carried out in Sections 1-7 below, whose headings are self explanatory.

Contents

1. Introduction ...... 1 2. Fibrewise suspensions ...... 4 3. Lusternik-Schnirelmann category ...... 6 4. Differential modules over differential algebras ...... 6 5. Sullivan models of fibrewise suspensions ...... 8 6. Topology of the functor Ext ...... 11 7. Applications to Poincaréduality ...... 13 Page 4 of 16 YVES FELIX,´ ANICETO MURILLO AND DANIEL TANRE´

References ...... 15

In this text, all spaces are compactly generated and of the homotopy type of a CW-complex. p By a ﬁbration F → E → B, we mean a map p whose homotopy ﬁbre is F .

2. Fibrewise suspensions The q-th suspension of a space X, ΣqX, is the pushout q q q−1 Σ X = (X × D ) ∪X×Sq−1 ({∗} × S ).

As usual, we denote by X+ the disjoint union of X and a point. Then, q q q Σ X+ = (X × D ) ∪X×Sq−1 ({∗} × D ). q q q In particular, we have Σ ∗ = ∗, Σ ∗+ = S , H˜∗(ΣX)= sH˜∗(X) and H˜∗(ΣX+)= sH∗(X). Following Crabb and James [3], a pointed fibration p: E → B is a fibration with a fixed section. A morphism of pointed fibrations is a morphism of fibrations compatible with the prescribed sections. To each fibration p: E → B, we associate a pointed fibration, p: E+ → B with E+ = E ∐ B. This notation is coherent with the notation X+ for a space X, the space X being considered as the total space of a fibration X →∗. With these notations, if p: E → B is a fibration of fibre F , E+ → B is a fibration of fibre F+.

Definition 1. Let q ≥ 1 and p: E → B be a fibration of fibre F . The q-th fibrewise suspension of p is the pushout q q q − −1 ΣBE = (E × D ) ∪E×Sq 1 (B × S ).

When we apply this to the pointed fibration p: E+ → B, we obtain the augmented q-th fibrewise suspension of the fibration p, q q q ΣBE+ = (E × D ) ∪E×Sq−1 (B × D ).

p Proposition 2.1. Let F → E → B be a ﬁbration. The ﬁbrewise suspension of p, ΣBE, is p p the homotopy pushout of B ←− E −→ B.

Proof. We deﬁne the ﬁbrewise cone over E as the homotopy pushout E ×{1} / E × [0, 1]

p g f / B ×{1} / CBE.

Then, if we denote by h: E ×{0} → CB E the composition of g with the inclusion of E ×{0} p×{0} h into E × [0, 1], ΣBE is the homotopy pushout of B ×{0} ←− E ×{0} −→ CB E. As we have q a ﬁbration CF → CBE → B, the projection q : CBE → B is a homotopy equivalence which precomposed with h gives the same homotopy pushout. As p = q ◦ h, the result follows.

The fibrewise suspension and the augmented fibrewise suspension are functorial and the cube lemma [13] implies that q / q / q / q / Σ F /ΣBE /B and Σ F+ /ΣBE+ /B FIBREWISE STABLE RATIONAL HOMOTOPY Page 5 of 16 are fibrations. The next result connects these two fibrewise constructions.

p Proposition 2.2. Let F → E → B be a ﬁbration and let q ≥ 1. Then there is a homotopy pushout

µq q−1 / Σq E B × S _ B

jq νq q / q B × D / ΣBE+, q q and the map νq : ΣBE → ΣBE+ is a map of ﬁbrations over B.

Proof. In the diagram below, the upper square and the global rectangle are homotopy pushouts, therefore the bottom square is a homotopy pushout as well. E × Sq−1 / E × Dq

q−1 / q B × S / ΣBE

q / q B × D / ΣBE+

Example 1. We describe the augmented q-th ﬁbrewise suspension of the trivial ﬁbration ∗ → X → X. The homotopy pushout X × Sq−1 / X × Dq

X / X × Sq q q tells us that ΣX X+ = X × S . Hence, the augmented q-th ﬁbrewise suspension of ∗ → X → X is the trivial ﬁbration Sq → X × Sq → X.

Example 2. q Consider the path ﬁbration, ΩX → PX → X. The spaces ΣX PX+ and the q homotopy ﬁbre of ΣX PX+ → X are the respective lower right hand corners of following homotopy pushouts PX × Sq−1 / PX × Dq ΩX × Sq−1 / ΩX

X / X ∨ Sq ∗ / ΩX ⋉ Sq. According to the cube lemma [13], taking homotopy fibres of the projection to X of the diagram on the left gives rise to the diagram on the right. Hence, the augmented q-th fibrewise suspension of the path fibration ΩX → PX → X is the fibration ΩX ⋉ Sq → X ∨ Sq → X. Page 6 of 16 YVES FELIX,´ ANICETO MURILLO AND DANIEL TANRE´

3. Lusternik-Schnirelmann category As an illustration of the use of fibrewise suspensions, we provide a characterization of the invariant Qrcat in terms of fibrewise suspensions of the Ganea fibrations. Recall that the LS category of a space X is the least integer n such that X can be covered by (n + 1) open sets contractible in X. For CW-complexes, an equivalent definition is given using Ganea fibrations pn : Gn(X) → X, with fibre in : Fn(X) → Gn(X), defined as follows: p0 is the path space fibration and pn+1 is the fibration associated to the extension of pn over 8 Gn(X) ∪in CFn(X) → X that sends CFn(X) to the basepoint. In [ ], Ganea proved that cat(X) ≤ n if, and only if, pn admits a section. For other equivalent definitions and basic properties of LS category, we refer to [2]. The invariant Qrcat was introduced in [15] as a lower bound for LS category in connection with the search for spaces satisfying the Ganea conjecture. For its definition consider the functor Qr =ΩrΣr and denote by Q˜r its fibrewise version. Then, by definition, Qrcat(X) ≤ n if the fibration r / r pñ / Q Fn(X) /Q˜ Gn(X) /X has a section.

r Proposition 3.1. If X is a CW-complex, then Q cat X ≤ n if, and only if, the map νr, introduced in Proposition 2.2, admits a retraction as pointed ﬁbration over X, i.e., if there exists a map of pointed ﬁbrations ϕ, r νr / r Σ Gn(X) o / Σ Gn(X)+ X I o______X II ϕ tt II tt II tt II tt I$ ytt X,

such that ϕ ◦ νr ≃ id.

Proof. In [18, (iv) of Proposition 10], Lucile Vandembroucq proves that Qrcat X ≤ n if, r r and only if, there exists a map φ: X × D → ΣX Gn(X) such that the diagram µ r−1 r / r X × S / Σ Gn(X) k5 X φ k k k k k k X × Dr / X

is commutative, where µr is deﬁned in Proposition 2.2. The result is now a direct consequence of Proposition 2.2.

4. Differential modules over differential algebras We first recall some definitions and basic properties of differential graded modules (dgm henceforth) over commutative differential graded algebras (cdga henceforth) which can be found, for instance, in [5]. Let lk be a commutative ring and (A, d), or A for short, be a cdga. We suppose that (A, d) is a cochain algebra which implies that the differential d has degree 1. A graded differential A-module is a differential graded lk-module, M, with a linear map of degree 0, A ⊗ M → M, a ⊗ m 7→ a.m, such that (a.b).m = a.(b.m), 1.m = m, d(a.m) = (da).m + (−1)|a|a.(dm), for any m ∈ M and a ∈ A of degree |a|. FIBREWISE STABLE RATIONAL HOMOTOPY Page 7 of 16

Let M and N be dgm’s over A. An A-linear map of degree j from M to N is a linear map f : M → N of degree j such that f(a.m) = (−1)j|a|a.f(m). (Observe that we do not require compatibility with the differentials.) If M,N are A-modules, we denote by homA(M,N) the graded A-module of A-linear maps from M to N, with the differential Df = df − (−1)|f|fd. j The submodule of A-linear maps of degree j is denoted homA(M,N). A morphism, ϕ: M → N, of A-modules is an A-linear map of degree 0 compatible with the differentials. The category A-Mod is the category whose objects are A-modules and morphisms the morphisms of A-modules. Two morphisms of A-modules, ϕ, ψ : M → N, are homotopic if there exists an A-linear map h: M → N, of degree −1, such that Dh = f − g. We denote this 0 relation by f ≃ g. A morphism of A-modules is therefore a cocycle in homA(M,N) and two morphisms are homotopic if, and only if, they are cohomologous in homA(M,N). An A-module M is semifree if it is isomorphic to A ⊗ V where V is a graded sum of free n−1 graded lk-modules, V = ⊕n≥0V(n), such that d(V(n)) ⊂⊕i=0 A ⊗ V(i), for any n ≥ 0. A semifree resolution of an A-module M is a semifree A-module P , together with a quasi- isomorphism ϕ: P ∼ /M .

Let M and N be two A-modules. By deﬁnition, ExtA(M,N)= H(homA(P,N)) in which P ∼ /M is a semifree resolution. This deﬁnition does not depend on the choice of P . A semifree extension of an A-module M is an A-module of the form M ⊕ (A ⊗ V ), where V is n−1 a free graded lk-module, V = ⊕n≥0V(n), with d(V(n)) ⊂ M ⊕ (A ⊗ (⊕i=0 V(i))), for any n ≥ 0. As an important property of semifree modules, for any morphism of A-modules, f : M → N, there exists a commutative diagram of morphisms of A-modules,

f M o 9/ N rr9 ∼ rr rr rr ϕ ( rr M ⊕ (A ⊗ V ) where M ⊕ (A ⊗ V ) is a semifree extension of M, and ϕ is a quasi-isomorphism. We construct now a cdga from a semifree A-module. Let (A ⊗ V, d) be a semifree A-module, the A-module (A ⊗ (Q ⊕ V ), d) is a cdga, with unit 1 ⊗ 1 ∈ A ⊗ Q, and with multiplication deﬁned by (A ⊗ V ) · (A ⊗ V )=0. This correspondence induces directly an isomorphism, for semifree modules (A ⊗ V, d) and (A ⊗ W, d),

∼ A homA−Mod(A ⊗ V, A ⊗ W ) = homcdga(A ⊗ (Q ⊕ V ), A ⊗ (Q ⊕ W )) . Here the left hand term denotes the set of morphisms of A-modules and the right hand term the set of cdga morphisms f, such that f(V ) ⊂ A ⊗ W and f|A = idA. This isomorphism is compatible with homotopy.

Proposition 4.1. Let f1,f2 ∈ homA−Mod(A ⊗ V, A ⊗ W ) with associated cdga morphisms A f 1, f 2 ∈ homcdga(A ⊗ (Q ⊕ V ), A ⊗ (Q ⊕ W )). Then,

f 1 ≃ f 2 as augmented A-cdga maps if, and only if, f1 ≃ f2 in A-mod.

As a consequence, we have an isomorphism

[A ⊗ V, A ⊗ W ]A−Mod =∼ [A ⊗ (Q ⊕ V ), A ⊗ (Q ⊕ W )]A, Page 8 of 16 YVES FELIX,´ ANICETO MURILLO AND DANIEL TANRE´

where the right hand side is the set of homotopy classes of augmented A-cdga maps, i.e., maps f whose restriction to A is the identity map and such that f(V ) ⊂ A ⊗ W . The homotopies are the identity on A and are compatible with the augmentations, see the path object described in the proof below.

Proof. To deﬁne homotopy in the category of A-modules and in the category of A-algebras we need path objects in those categories. For the semifree A-module (A ⊗ W, d), we construct the semifree A-module

(A ⊗ (W1 ⊕ W2 ⊕ sW ),D) , n n−1 where A ⊗ W1 and A ⊗ W2 are copies of A ⊗ W , (sW ) = W , D(w1)= d(w1)+ sw, D(w2)= dw2 − sw, D(sw)= −S(dw) where S : A ⊗ W → A ⊗ W is the A-linear map (of degree one) generated by w 7→ sw. The projections p1,p2 : (A ⊗ (W1 ⊕ W2 ⊕ sW ),D) → (A ⊗ W, d) deﬁned by p1(w1)= w, p1(w2)=0, p2(w1)=0, p2(w2)= w, p1(sw)= p2(sw)=0, are quasi-isomorphisms. These two maps together with the injection j : (A ⊗ W, d) → (A ⊗ (W1 ⊕ W2 ⊗ sW ),D), deﬁned by j(w)= w1 + w2, are a path object in the category of A-modules. Therefore two morphisms of A-modules f,g : (A ⊗ V, d) → (A ⊗ W, d) are homotopic if, and only if, there is a map

Φ: (A ⊗ V, d) → (A ⊗ (W1 ⊕ W2 ⊕ sW ),D)

such that f = p1 ◦ Φ and g = p2 ◦ Φ. Now we do essentially the same with the cdga (A ⊗ (Q ⊕ W ), d). We construct the A-cdga

(A ⊗ (Q ⊕ W1 ⊕ W2 ⊕ sW ),D) ,

with the same diﬀerential as above and with trivial product on A ⊗ (W1 ⊕ W2 ⊕ sW ). The projections p1,p2 : (A ⊗ (Q ⊕ W1 ⊕ W2 ⊕ sW ),D) → (A ⊗ (Q ⊕ W ), d) are quasi-isomorphisms of A-cdga’s. Therefore two maps of A-cdga’s, f,g : (A ⊗ (Q ⊕ V ), d) → (A ⊗ (Q ⊕ W ), d), are A-homotopic if, and only if, there is a map of A-cdga’s

Φ: (A ⊗ (Q ⊕ V ), d) → (A ⊗ (Q ⊕ W1 ⊕ W2 ⊕ sW ),D)

such that f = p1 ◦ Φ, g = p2 ◦ Φ and Φ(V ) ⊂ A ⊗ (W1 ⊗ W2 ⊗ sW ). This directly implies the result.

5. Sullivan models of fibrewise suspensions This section is concerned with Sullivan models for which we refer to [5], [7] or [17]. We describe models for the augmented fibrewise fibrations associated to a given fibration. The (Sullivan) minimal model of a connected space, X, is a cdga (∧V, d), which is unique up to isomorphism and whose cohomology is the rational cohomology of X. When X is a nilpotent space, with finite Betti numbers, this cdga determines the rational homotopy type of X. The algebra ∧V is the free commutative graded algebra generated by the graded vector space V . The differential d(v) of any element v of V is a polynomial in ∧V with no linear term. Moreover V admits a basis vi indexed by a well ordered set I such that d(vi) ∈ ∧(vj ,j

p Proposition 5.1. Let F → E → B be a rational fibration modeled by (A, d) → (A ⊗ ∧V,D). (i) Suppose that the fibration p has a section modeled by the projection ρ: (A ⊗ ∧V,D) → (A, d) defined by ρ(V )=0. Then a model for the fibrewise suspension fibration ΣB E → B is (A, d) −→ (A ⊗ (Q ⊕ s ∧+ V ),D) where the product of two elements in s ∧+ V is zero and, if Φ ∈ ∧+V has for differential + DΦ= ai ⊗ Φi, with Φi ∈ ∧ V , then Pi D(sΦ) = (−1)|ai|+1a ⊗ sΦ . X i i i

(ii) A model for the ﬁbrewise augmented suspension ΣB E+ → B is (A, d) −→ (A ⊗ (Q ⊕ s ∧ V ),D) where the product of two elements in s ∧ V is zero and, if Φ ∈ ∧V has for diﬀerential DΦ= ai ⊗ Φi, with Φi ∈ ∧V , then Pi D(sΦ) = (−1)|ai|+1a ⊗ sΦ . X i i i

Proof. We begin by considering the model of S0 = ∗∐∗ given by Qa ⊕ Qb, with a and b in degree 0, a2 = a, b2 = b, ab = 0; the unit of this cdga is a + b. Hence, if we set t = a − b, the previous cdga is isomorphic to ∧(t)/(t2 − 1) = Q ⊕ Qt. → (Then, a model of the inclusion S0 ֒→ D1 is given by the canonical surjection ϕ: ∧ (t,dt ∧(t)/(t2 − 1), with ϕ(dt) = 0. As this map is surjective, Q ⊕ ker ϕ is a model of S1, the coﬁbre of S0 → D1, see [5, Page 169] or [9, Proposition 5.18]. Write K = ker ϕ and observe that this kernel is the ideal generated by t2 − 1 and dt. The injection of the sub-cdga Q ⊕ Qdt in Q ⊕ K is a quasi-isomorphism, which implies that Q ⊕ Qdt is a model of S1, as expected.

We now prove (i). By Deﬁnition, ΣBE is the following homotopy pushout id×ι E × S0 / E × D1

p×id 0 / B × S / ΣBE. By the above observations, the cdga morphisms,

j id⊗ϕ A ⊗ ∧(t)/(t2 − 1) / (A ⊗ ∧V ) ⊗ ∧(t)/(t2 − 1) o (A ⊗ ∧V ) ⊗ ∧(t,dt) are models of id × ι and p × id. As id ⊗ ϕ is surjective, the model of ΣB E is the pullback M of these morphisms, [5, Page 169] or [9, Proposition 5.18], that is M = Q ⊕ ker((id ⊗ ϕ) − j) where (id ⊗ ϕ) − j): ((A ⊗ ∧V ) ⊗ ∧(t,dt)) ⊕ A ⊗ ∧(t)/(t2 − 1) → (A ⊗ ∧V ) ⊗ ∧(t)/(t2 − 1). As graded vector space, we have M = Q ⊕ ker (id ⊗ ϕ) ⊕ (A ⊗ ∧(t)/(t2 − 1)). Page 10 of 16 YVES FELIX,´ ANICETO MURILLO AND DANIEL TANRE´

By hypothesis, A ⊗ Q ⊕ (∧+V ⊗ dt) ,D is a diﬀerential subalgebra of M and the quotient Z of M by this subalgebra is the centre of a short exact sequence of diﬀerential vector spaces, 0 → A ⊗ ∧+V ⊗ I → Z → A ⊗ ∧+(t,dt) → 0, where I is the ideal of ∧(t,dt) generated by t2 − 1 and t dt. As I and ∧+(t,dt) are acyclic, the quotient Z is also acyclic and we have a quasi-isomorphism between the pullback M and (A ⊗ (Q ⊕ s ∧+ V ),D) =∼ A ⊗ Q ⊕ (∧+V ⊗ dt) ,D . (ii) follows from (i) because a model of E+ → B is (A, d) → (A ⊗ (Q ⊕ ∧V ),D).

The above constructions being natural, a straightforward computation describes the models for the augmented ﬁbrewise suspension of a map between ﬁbrations.

Proposition 5.2. Let p: E → B and p′ : E′ → B be ﬁbrations over B modeled by the cdga’s A → A ⊗ ∧W and A → A ⊗ ∧V respectively. Let f : E → E′ be a map over B modeled by ξ : A ⊗ ∧V → A ⊗ ∧W . Then, with the notation of Proposition 5.1, we get the following models of ΣBf and ΣBf+. (i) If p and p′ are pointed ﬁbrations and f is a pointed map, then a model for the map ΣBf is given by + + ΣAξ : A ⊗ (Q ⊕ s ∧ V ) → A ⊗ (Q ⊕ s ∧ W )

|ai| + where ΣAξ(sΦ) = i(−1) ai ⊗ sΦi, if Φ ∈ ∧ V and ξ(Φ) = a ⊗ 1+ i ai ⊗ Φi, with + P P Φi ∈ ∧ V . (ii) A model for the map ΣBf+ is given by

(ΣAξ)+ : A ⊗ (Q ⊕ s ∧ V ) → A ⊗ (Q ⊕ s ∧ W )

|ai| deﬁned by (ΣAξ)+(sΦ) = (−1) ai ⊗ sΦi, if Φ ∈ ∧V and ξ(Φ) = ai ⊗ Φi, with Pi Pi Φi ∈ ∧V .

Example 3. The Hopf fibration. We apply Proposition 5.1 to get a model of the augmented fibrewise fibration associated to the Hopf fibration S3 → S2. The model of the Hopf fibration, is given by (A ⊗ ∧z,D) , in which A = (∧x/x2, 0), |x| =2, is a model for S2, |z| = 1 and Dz = x. Then, a model for ΣBE+ is given by (A ⊗ (Q ⊕ sQ ⊕ szQ),D) , D(sz)= x ⊗ s1 . A direct computation shows that this cdga has reduced homology in degree 1, 2 and 4, with respective representative cocycles 1 ⊗ s1, x ⊗ 1 and x ⊗ sz. To determine the corresponding CW-structure, we compute its minimal model in low degrees, ϕ: (∧(u,v,w,t,...), d) → (A ⊗ (Q ⊕ sQ ⊕ szQ),D) , with ϕ(u)=1 ⊗ s1, ϕ(v)= x ⊗ 1, ϕ(w)=1 ⊗ sz, ϕ(t)=0, du = dv = 0, dw = uv, dt = wu. The cohomology classes are represented by the cocycles 1,u,v and wv. Therefore the homology 1 2 classes in degree 1 and 2 are images by the Hurewicz map of homotopy classes, S and Sa, 1 2 2 and the iterated Whitehead bracket [S , [Sa,Sa]] is killed. Therefore the augmented fibrewise 2 fibration, ΣBE+ → S , has the rational homotopy type of the projection 2 1 4 2 1 2 2 Sa ∨ S ∪[S ,[Sa,Sa]] e → Sa, FIBREWISE STABLE RATIONAL HOMOTOPY Page 11 of 16

1 that is trivial on the cells in degree 1 and 4. The injection of the ﬁbre, ΣS+ → ΣBE+, has the rational homotopy type of the injection

1 2 2 1 4 1 2 2 S ∨ Sb → Sa ∨ S ∪[S ,[Sa,Sa]] e , 2 1 2 where the restriction to Sb is the Whitehead bracket [S ,Sa]. Observe also that, for reason of degree, the fibrewise suspension of the Hopf fibration is the trivial fibration S2 → S2 × S2 → S2.

Proposition 5.3. For a ﬁbration p: E → B, the holonomy operation of H∗(ΩB; Q) on H∗(F ; Q) is completely determined by the ﬁbrewise stable rational homotopy type of E.

Proof. Denote by (∧V, d) → (∧V ⊗ ∧W, D) a relative minimal model for p, with (∧V, d) a minimal model for B. Then the holonomy operation is completely determined by the component D1 : W → V ⊗ ∧W of D [5]. This part is preserved by ﬁbrewise suspension (cf. Proposition 5.1). This implies the result.

Example 4. The ﬁbrewise rational homotopy type of E contains much more information than the holonomy operation. Consider the following example

3 3 3 3 3 8 9 E = T (S ,S ,S ) ∪ 3 3 (S × S × S ) ∪ω−id e h a b c Sa×Sb a b i S8 where T (X,Y,Z) denotes the fat wedge of the spaces X, Y and Z and where ω ∈ 3 3 3 3 3 3 π8(T (Sa,Sb ,Sc )) is the homotopy class of the attaching map of the top cell in Sa × Sb × Sc . 3 3 The projection on Sa × Sb induces a homotopy fibration whose homotopy fibre has the 3 8 rational homotopy type of Sc ∨ S . A semifree model of E is given by the cdga (∧(a,b) ⊗ Q(1,c,u),D) , with Du = abc, |a| = |b| = |c| = 3, |u| = 8, cu = u2 =0. By [5], the holonomy operation is trivial. However the augmented q-th fibrewise suspension is

q 3 3 3+q 3 3 8+q 9+q Σ E+ = T (S ,S ,S ) ∪ 3 3 (S × S × S ) ∪ω−id e , B h a b c Sa×Sb a b i S8+q 3 3 which is not a trivial ﬁbration over Sa × Sb .

6. Topology of the functor Ext Unless explicitly stated, the spaces of this section are rational and nilpotent CW-complexes; singular cochain and cohomology algebras are taken over the rationals and Sq denotes the rationalization of the q-sphere.

Let p: E → B and p′ : E′ → B be ﬁbrations. We denote by [E, E′] the set of homotopy ′ B ′ ′ q q ′ B classes of maps from E to E over B and, for q, q > 0, by [Σ E+, Σ E ] the set of homotopy ′ B B + B q q ′ classes of pointed maps over B from ΣBE+ to ΣB E+.

Definition 2. Let p: E → B and p′ : E′ → B be ﬁbrations. The set of stable ﬁbrewise homotopy classes of maps from E to E′ is the limit

′ r q q+r ′ B {E, E } = lim [Σ E+, Σ E ] . B q B B + B Page 12 of 16 YVES FELIX,´ ANICETO MURILLO AND DANIEL TANRE´

′ r q q ′ B More precisely, {E, E }B is the quotient of the union ∪q [ΣBE+, ΣBE+]B by the equivalence relation induced by the relation fRg if and only if Σf ≃ Σg.

Proof of Theorem 1.1. Denote by (A ⊗ ∧V,D) and (A ⊗ ∧V ′,D) the relative cdga models of the ﬁbrations p and p′, respectively. When p and p′ are rational ﬁbrations, there is a bijection between the sets of homotopy classes

∼ ′ = / ′ [E, E ]B /[A ⊗ ∧V , A ⊗ ∧V ], where the right hand term is the set of A-homotopy classes between A-cdga’s. Recall from Section 4 that a morphism, f : M → N, of A-modules is a cocycle in homA(M,N) and that two morphisms of A-modules are homotopic if, and only if, they are homologous in homA(M,N). We therefore get a map

′ 0 ′ ϕ: [A ⊗ ∧V , A ⊗ ∧V ]A → ExtA(A ⊗ ∧V , A ⊗ ∧V ) . As seen in Proposition 5.1, a model for the augmented q-th ﬁbrewise suspension of p is A ⊗ (Q ⊕ sq ∧ V ). Denote by A ⊗ ∧V (q) (respectively A ⊗ ∧V ′(q)) a relative minimal model for A ⊗ (Q ⊕ sq ∧ V ) (resp. A ⊗ (Q ⊕ sq ∧ V ′)). Observe that these models are augmented by V (q) 7→ 0 and V ′(q) 7→ 0. Taking augmented ﬁbrewise suspensions gives the following diagram

ϕ ′ / 0 ′ [A ⊗ ∧V , A ⊗ ∧V ] / ExtA(A ⊗ ∧V , A ⊗ ∧V )

Σ =∼ ϕ ′ 1 / 0 ′ [A ⊗ ∧V (1), A ⊗ ∧V (1)]A / ExtA(A ⊗ s ∧ V , A ⊗ s ∧ V )

Σ =∼ ϕ ′ 2 / 0 2 ′ 2 [A ⊗ ∧V (2), A ⊗ ∧V (2)]A / ExtA(A ⊗ s ∧ V , A ⊗ s ∧ V )

Σ =∼ . . . . . . The left hand term at the top is the set of homotopy classes of A-cdga maps. The other terms of the left column are the sets [−, −]A of homotopy classes of augmented A-cdga’s, as in Proposition 4.1 and its proof. To any A-linear map, f : A ⊗ ∧V ′ → A ⊗ ∧V , we associate an A-linear map, g : A ⊗ s ∧ V ′ → A ⊗ s ∧ V , deﬁned by g(sw)= sf(w). This association being clearly bijective, the right vertical maps are isomorphisms. By taking the limit, the morphisms ϕq give a map ′ 0 0 ′ Φ: {E, E }B → ExtA(A ⊗ ∧V , A ⊗ ∧V ).

0 q ′ q Let q ≥ 0. Recall now that the set ExtA(A ⊗ s (∧V ), A ⊗ s (∧V )) is the set of homotopy q ′ q classes [A ⊗ s ∧ V , A ⊗ s ∧ V ]A−Mod, which is isomorphic to the set of homotopy classes q ′ q of augmented A-cdga’s [A ⊗ (Q ⊕ s ∧ V ), A ⊗ (Q ⊕ s ∧ V )]A. We therefore have a section jq of ϕq whose image is given by the homotopy classes of morphisms from (A ⊗ ∧V (q),D) to (A ⊗ (Q ⊕ sq ∧ V ′),D) that factor up to homotopy through (A ⊗ (Q ⊕ sq ∧ V ),D). This implies the surjectivity of Φ. Moreover, by construction, the images of the suspension Σ belong to the image of the corresponding section j. Therefore, if Φ([f]) = Φ([g]), then [Σf]=[Σg]. This proves the injectivity of Φ. ′ r ∼ r ∗ ′ ∗ The general case {E, E }B = ExtC∗(B;Q)(C (E ; Q), C (E; Q)) follows from a similar argu- ment. FIBREWISE STABLE RATIONAL HOMOTOPY Page 13 of 16

Remark 1. Rationally, any suspension, Σf : ΣqX → ΣqY , is the (q − 1)-th suspension of ′ r a map from ΣX to ΣY . Therefore, each element of {E, E }B can be represented by a pointed r+1 ′ r+1 ′ map over B, f : ΣBE+ → ΣB E . Moreover two maps f,g : ΣBE+ → ΣB E deﬁne the same ′ r elements in {E, E }B if, and only if, ΣBf ≃ ΣBg. This is a consequence of the implication Φ([f]) = Φ([g]) ⇒ [Σf]=[Σg], established in the previous proof.

Example 5. Let F → E → B be a fibration. The well known isomorphism, ∗ ∗ ∼ ExtC∗(B)(C (E), Q) = H∗(F ; Q), can be revisited in light of Theorem 1.1 as follows: n ∗ q q+n B q q+n q+n ∗ Q ∼ ∼ ExtC (B)(C (E), ) = lim B ∨ S , ΣB E+ B = lim S , Σ F+ = lim πq(Σ F+) →q →q →q = Hn(F ; Q) . In particular, for any simply connected space X, we have: ∼ ∗ ∼ ∗ H∗(ΩX; Q) = ExtC∗(X;Q)(Q, Q) = {PX,PX}X . Let (P, d) → (Q, 0) be a resolution of (Q, 0) as C∗(X; Q)-module. The multiplication in H∗(ΩX; Q) is then obtained by the composition of morphisms in HomC∗(X;Q)(P, P ). This ∗ multiplication is thus given by composition of morphisms in {PX,PX}X. ∗ ∗ 1 The computation of {PX,PX}X = ExtC∗(X;Q)(Q, Q) is difficult. By [ , §4], there is a space X and a sequence of integers an such that, fixing in advance a first order axiomatization of set n theory, the statement dim {PX,PX}X = an is undecidable.

7. Applications to Poincaréduality This section is devoted to the study of the rational vector spaces ∗ r r q r+q X X,PX = ⊕r≥0 X,PX , {X,PX}X = lim [X × S ,X ∨ S ]X X X q→+∞ and their connection with Poincaréduality. Here all spaces are rational and nilpotent CW- complexes, with finite Betti numbers. Singular cochain and cohomology algebras are taken over the rationals and Sq denotes the rationalization of the q-sphere. Recall that the augmented q-th fibrewise suspension of ∗ → X → X is the trivial fibration Sq → X × Sq → X and that the augmented q-th fibrewise suspension of the path fibration ΩX → PX → X is the fibration ΩX ⋉ Sq−1 → X ∨ Sq → X. The application of Theorem 1.1 to these cases gives directly the next statement.

Proposition 7.1. For any q ∈ Z, we have the following isomorphisms, q ∗ q n n+q X Ext ∗ Q, C (X) =∼ X,PX = lim[X × S ,X ∨ S ] C (X) X n X and q q n n+q X Ext ∗ (Q, Q) =∼ PX,PX = lim[X ∨ S ,X ∨ S ] . C (X) X n X

∗ Now recall that, given f : X × Y → Z, a ∈ H∗(Y ) and v ∈ H (Z), the slant product H∗(f)(v)/a ∈ H∗(X) is the class deﬁned by hH∗(f)(v)/a,ui = hH∗(f)(v),a × ui, u ∈ H∗(X). When f : X × Sq → X ∨ Sq+n is a map over X, and a, v are the fundamental classes of Sq and Sn+q respectively, the slant product deﬁnes a map ∗ θ : X,PX −→ H∗(X) . X Page 14 of 16 YVES FELIX,´ ANICETO MURILLO AND DANIEL TANRE´

In terms of models, θ can be described as follows: let (ΛV, d) and (ΛW, d) be models of X and X ∨ Sq+n respectively, and denote by y a generator of W q+n representing the sphere Sq+n. Recall that a model for Sq is the cdga (∧z/z2, 0) with z in degree q. Then, a map f : X × Sq → X ∨ Sq+n is modeled by a morphism ϕ: (ΛW, d) → (ΛV, d) ⊗ (∧(z)/z2, 0) with ϕ(y)= zΦ+Ψ. Then θ(f)=Φ.

On the other hand, the evaluation map

∗ ∗ ∗ ev : Ext ∗ Q, C (X) → H (X) C (X) is a natural linear map which, among other interpretations [4, 6, 14], can be seen as the morphism ∗ ∗ ∗ ∗ ∗ Ext ∗ Q, C (X) → Ext Q, C (X) =∼ H (X) C (X) Q induced by the inclusion Q ֒→ C∗(X). With these remarks, the proof of Theorem 1.3 reduces to the proof of the following statement.

Proposition 7.2. The following diagram commutes:

∗ ∗ ∗ / ∗ Q X,PX X ∼ ExtC (X) , C (X) LL = m LL mmm LLL mmm θ LL mmm ev L& vmm H∗(X), where the top map is the isomorphism exhibited in Proposition 7.1.

Proof. On one hand, θ is the composition ∗ ∗ α X,PX → X,PX −→ H∗(X), X point obtaining via the collapsing X ∨ Sn+q → Sn+q, and where α associates to any map f : X × q q+m ∗ S → S , the cohomology class H (f)(aq)/aq+m, where aq,aq+m are the fundamental classes of Sq and Sq+m respectively. On the other hand, Theorem 1.1 implies immediately the existence of an isomorphism ∗ X,PX =∼ Ext∗ Q, C∗(X) point Q which is easily seen to ﬁt in the following commutative diagram,

∗ ∗ / X,PX X,PX X point NN NNNα ∼= ∼= NN NNN N& ∗ ∗ / ∗ ∗ / ∗ Ext ∗ Q, C (X) / ExtQ Q, C (X) / H (X) C (X) =∼

Proof of Theorem 1.4. According to [4, Theorem 3.1], X is a Poincar´eduality complex if, ∗ ∗ and only if, Ext ∗ Q; C (X; Q) is a ﬁnite dimensional vector space, in which case it is C (X;Q) of dimension one and concentrated in degree N = dim X.

Proof of Theorem 1.5. This follows directly from [4, Proposition 4.3] and [14, Theorem A], together with Theorem 1.1. FIBREWISE STABLE RATIONAL HOMOTOPY Page 15 of 16

Remark 2. Observe that, whenever X is a Poincar´ecomplex, the only non trivial stable class given by Theorem 1.4 is represented by the rationalization of X × Sq → X × Sq/X = X ∨ ΣqX → X ∨ Σq(X/X

Example 6. Let X = CP ∞. The rational vector space ∞ ∞ ∞ r ∞ q ∞ r+q CP C C ∞ C C ∞ ( P )Q, (P P )Q CP = lim [ ( P × S )Q , ( P ∨ S )Q ]CP Q q→+∞ is zero except for r = −1, in which case, it is a one dimensional vector space. ∗ ∞ In fact Ext ∗ ∞ Q, C (CP ; Q) has dimension one and is concentrated in degree −1. C (CP ;Q) To describe a generator, we remark ﬁrst that a model for CP ∞ ∨ Sq−1 is given by (∧(a,x,y)/(x2,y2, xy), d) with |a| = 2, |x| = q, |y| = q − 1, d(a)= d(y)=0, d(x)= ay. The canonical projection on 2 ∞ q ∞ q−1 (∧a ⊗ ∧x/x , 0) can be realized as a map f : (CP × S )Q → (CP ∨ S )Q. The image of Sq is the Whitehead bracket [S2,Sq−1], where S2 denotes the 2-cell of CP ∞. Observe that θ(f)=0.

Example 7. Let X = CP ∞ × Y with Y a simply connected Poincaréduality complex of dimension N. Then we consider the composition ∞ q ϕ1 ∞ q+N ∞ ∞ q+N CP × (Y × S ) → CP × (Y ∨ S ) = (CP × Y ) ∪CP ∞ (CP × S ) ϕ2 ∞ ∞ q+N−1 ∞ q+N−1 → (CP × Y ) ∪CP ∞ (CP ∨ S ) → (CP × Y ) ∨ S . q q+N Here ϕ1 is the map Y × S → Y ∨ S defined by the Poincaréduality structure on Y and ϕ2 is the map CP ∞ × Sq → CP ∞ ∨ Sq+N−1 defined above. The composition is, up to a multiple, the only non trivial rational stable map from (CP ∞ × Y ) × Sq to (CP ∞ × Y ) ∨ Sq+N−1.

References 1. David Anick, Diophantine equations, Hilbert series, and undecidable spaces, Annals of Math., 122 (1985), 87–112. 2. Octav Cornea, Gregory Lupton, John Oprea and Daniel Tanré, Lusternik-Schnirelmann Category, Math. Surveys and Monographs 103, AMS, 2003. 3. Michael Crabb and Ioan James, Fibrewise Homotopy Theory, Springer Monographs in Mathematics, 1998 4. Yves Félix, Stephen Halperin and Jean-Claude. Thomas, Gorenstein spaces, Advances in Math., 71(1) (1989), 92–112. 5. Yves Félix, Stephen Halperin and Jean-Claude Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics 205, Springer, 2000. 6. Yves Félix, Jean-Michel Lemaire and Jean-Claude Thomas, L’application d’évaluation, les groupes de Gottlieb duaux et les cellules terminales, Journal of Pure and Appl. Algebra, 91 (1994), 143–164. 7. Yves Félix, John Oprea and Daniel Tanré, Algebraic Models in Geometry, Oxford Graduate Texts in Mathematics 17, Oxford University Press, 2008. 8. Tudor Ganea, Lusternik-Schnirelmann category and strong category, Illinois J. Math., 11 (1967), 417–427. 9. Stephen Halperin, Lecture on minimal models, Mémoires de la SociétéMathématique de France, Nouvelle Série 9-10, 1983. 10. Kathryn Hess, A proof of Ganea’s conjecture for rational spaces, Topology, 30 (1991), 205–214. 11. Barry Jessup, Rational L.-S. category and a conjecture of Ganea, Journal of Pure and Applied Algebra, 65 (1990), 57–67. 12. John Klein, Embedding, compression and fibrewise homotopy theory, Algebraic and Geometric Topology, 2 (2002), 311–336. 13. Michael Mather, Pull-backs in homotopy theory, Canad. J. Math., 28(2), (1976), 225–263. 14. Aniceto Murillo, On the evaluation map, Trans. Amer. Math. Soc., 339 (1993), 611–622. 15. Hans Scheerer, Donald Stanley and Daniel Tanré, Fibrewise construction applied to Lusternik-Schni- relmann category, Israel J. of Math., 131 (2002), 333-359. 16. Hans Scheerer and Manfred Stelzer, Fibrewise infinite symmetric products and M-category, Bulletin of the Korean Mathematical Society, 36 (1999), 671–682. Page 16 of 16 FIBREWISE STABLE RATIONAL HOMOTOPY

17. Daniel Tanr´e, Homotopie rationnelle: mod`eles de Chen, Quillen, Sullivan, Lecture Notes in Mathematics 1025, Springer, 1983. 18. Lucile Vandembroucq, Fibrewise suspension and Lusternik-Schnirelmann category, Topology, 41 (2002), 1239-1258.

Yves Félix Aniceto Murillo Département de Mathématiques Departamento de Algebra,´ Geometr´ıa y UniversitéCatholique de Louvain Topolog´ıa 2, Chemin du Cyclotron Universidad de Málaga 1348 Louvain-La-Neuve Ap. 59 Belgium 29080-Málaga España [email protected] [email protected]

Daniel Tanré Département de Mathématiques UMR 8524 Universitédes Sciences et Technologies de Lille 59655 Villeneuve d’Ascq Cedex France [email protected]