Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Fibrewise Stable Rational Homotopy 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 Poincar´ecomplex 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 Poincar´eduality 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 equiva- lent, 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 Poincar´eduality 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 Poincar´eduality ........... 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 ).