NILPOTENT SUBGROUPS OF THE GROUP OF FIBRE HOMOTOPY EQUIVALENCES

Let ξ = (E, p, B, F ) be a Hurewicz fibration. In this paper we study the space LG(ξ) consisting of fibre homotopy self equivalences of ξ inducing by restriction to the fibre a self homotopy equivalence of F belonging to the group G. We give in particular conditions implying that π1(LG(ξ)) is finitely generated or that L1(ξ) has the same rational homotopy type as aut1(F ). Let ξ = (E, p,B, F ) be a Hurewicz fibration where B and F are compactly generated spaces. The set of free (not necessarily fibre) homotopy classes of free fibre homotopy equivalences of ξ into itself is a group L(ξ), for the multiplication induced by the composition of maps. Recall that a fibre homotopy equivalence f : E → E induces an homotopy equivalence of p−1(b) for each b ∈ B (A theorem of Dold ([4, Theorem 6.3]) asserts that the converse is true if B is a CW complex). There exists thus a natural map

Let ξ = (E, p, B, F ) be a Hurewicz fibration where B and F are compactly generated spaces.The set of free (not necessarily fibre) homotopy classes of free fibre homotopy equivalences of ξ into itself is a group L(ξ), for the multiplication induced by the composition of maps.
Recall that a fibre homotopy equivalence f : E → E induces an homotopy equivalence of p −1 (b) for each b ∈ B (A theorem of Dold ([4, Theorem 6.3]) asserts that the converse is true if B is a CW complex).There exists thus a natural map where Aut F denotes the group of free homotopy classes of free homotopy equivalences of the space F into itself.
Our purpose in this paper is the study of the groups L G (ξ) = R −1 (G) and the spaces L G (ξ) where G is some subgroup of Aut F .Here aut X is the monoid of free homotopy equivalences of the space X into itself, aut G X is the submonoid of aut X consisting of the path components belonging to G, and L G (ξ) is the space of fibre homotopy self-equivalences of ξ inducing by restriction to the fibre an element of aut G F : When G is reduced to the identity {1}, we obtain the (connected) monoid aut 1 F of self-equivalences homotopic to the identity, and the monoid L 1 (ξ) of fibre self-equivalences of ξ inducing by restriction to the fibre a map homotopic to the identity.
The monoids L G (ξ) and aut G F are H-spaces, so that all their components have the same homotopy type.The study of the homotopy type of L G (ξ) is therefore reduced to the consideration of (a) the map π 0 (R) : Our main problems can be stated as follows : We first show that the group L 1 (ξ) and the groups π i (L 1 (ξ)), i ≥ 1, are finitely generated groups when the base B is a simply connected finite CW complex and the fibre F has the homotopy type of a simply connected finite type CW complex.In the particular case the base is a sphere, the result is more precise.We have indeed : then there exists an exact sequence of groups
(2) G α is the stabilizer of {α} in G for the natural action of G on In case G = Aut X, this result has been obtained by K. Tsukiyama ( [21]), as a corollary of a result of D. Gottlieb ([9]).Theorem 1 is obtained in a similar way from a slight modification of the quoted result of D. Gottlieb.
The interest of the above generalization of Tsukiyama's result lies in Theorem 2. Under the hypothesis of Theorem 1, if we suppose that F is a nilpotent space and that G acts unipotently on each Theorem 2 follows from Theorem 1 and Theorem 3.3 of ( [6]).Indeed, Theorem 3.4 of ( [6]) states that under our conditions the group G is nilpotent.
As a consequence of Theorem 2, we obtain after 0-localization the exact sequence where L G (ξ) and G α respectively denote the Malcev completions of the nilpotent groups L G (ξ) and G α .
Our next result gives a complete description of this exact sequence in terms of a Sullivan model of F (see ( [20], [11]) for basic notions in rational homotopy theory).
Let (∧X, d) be a minimal model for F with a fixed K.S. basis (x i ) i∈I .A derivation θ of (∧X, d) is locally nilpotent (rel.(x i )) if we have Denote by Der * ∧X the graded Lie algebra of derivations of (∧X, d).This is a Z-graded Lie algebra.The differential D = [d, −] makes Der * ∧X into a graded differential Lie algebra.We define the sub differential Lie algebra L * by : is the subspace of Der 0 (∧X) consisting of cycles which are locally nilpotent with respect to the fixed K.S. basis.
Theorem 3. Let ξ = (E, p, S n , F ) be a unipotent fibration with fibre a nilpotent space F , and let G be a maximal subgroup of Aut F acting unipotently on H * (F ; Z).If G is torsion free, then we have the exact sequence Note that the torsion free hypothesis on G is not difficult to satisfy.For instance, if X is a rational space, then Aut X is a torsion free group ([3, Theorem 2.5]).
On the other hand, if X is a finite type virtually nilpotent CW complex, then Aut X is finitely generated ( [5]).
Using rational homotopy, we can make precise the structure of L G (ξ) in two interesting cases.
It is well known that fibrations ξ with fibre an homogeneous space K/H with rank K = rank H have special properties.We know that the Serre spectral sequence of ξ with rational coefficients collapses at the E 2 -term.Here we show that the space of self-equivalences of ξ is very small.More precisely, Theorem 4. Let ξ : (E, p, B, F ) be a fibration where all spaces are simply connected and of the homotopy type of finite CW complexes.We suppose that F is an homogeneous space, F = K/H with K and H compact connected Lie groups of the same rank, and that H 2n+1 (B; Z) is a finite group for n ≥ 0. Let G be a maximal subgroup of Aut F acting unipotently on H * (F ; Z).Then, (a) the group L G (ξ) is a finite group.(b) the space L 1 (ξ) is a connected finite dimension H-space, and for n > 1, we have Remark that (a) means that two self-equivalences of ξ inducing homotopic restrictions to the fibre F localized at 0 are already homotopic, after localization at 0. In a similar way, we obtain Theorem 5. Let ξ : (E, p, B, F ) be a fibration where all spaces are simply connected and of the homotopy type of finite CW complexes.We suppose that there exists an integer n such that π q (F ) is finite for q > n and Hq (B; Z) is finite for q ≤ n.Let G be a maximal subgroup of Aut F acting unipotently on H * (F ; Z).Then,

Proof of Theorem 1
We consider the fibre sequence aut where e is the evaluation map.Taking the classifying space of the monoids aut • X and aut X, we get a fibration sequence (up to homotopy) which is universal for Hurewicz fibrations with fibre X, ([6, Proposition 4.1]).
By analogy with the theory of fibre bundles, we consider Aut F as the "structural group" of a Hurewicz fibration ξ = (E, p, B, F ) and we shall say that the structural group of ξ can be reduced to This is only a useful analogy because the classifying map does not factor at all through the classifying space B G .In fact we can form the monoid aut G F of self-equivalences of F whose homotopy classes belong to G.In case of a G-reduction the classifying map k factors through the space We denote by L(B, B aut G F ; k) the component of k in the space of maps from B to B aut G F and by Following the lines of the proof given by D. Gottlieb in the case B aut F ([9, Theorem 1]), we obtain Proposition 1.Let F → E → B be a fibration whose base is a CW complex and with classifying map k.If Φ is defined as above, then : ( This implies immediately : In the particular case when B = { * }, we have a fibration with fibre aut G F .Therefore we recover Corollary 2. If F is compactly generated, then Corollary 3. If B is a simply connected finite CW complex and F has the homotopy type of a simply connected finite type CW complex, then the groups π i (L 1 (ξ)), i ≥ 1, are finitely generated.
On the other hand, denoting by M the Sullivan minimal model of F , we have a sequence of group isomorphisms π n (aut [12, 3.11]).As π n (aut 1 (M )) is finitely generated, the same is true for π n (aut 1 (F )) for n ≥ 1.We now make use of the Federer spectral sequence ( [7]) converging to π * (L(B, B aut1 F , k)).It is easy to see that E 2 p,q = H q (B, π p+q (B aut1 F )) is finitely generated abelian so that E ∞ p,q is finitely generated abelian.Since an extension of finitely generated abelian groups is a finitely generated abelian group, the groups π n (L(B, B aut1 F , k)) are finitely generated.

Consider now the evaluation map
This is a Hurewicz fibration and the fibre is the space of based maps L • (S n , B aut G F ).It results from ([22, Theorem 3.2]) that the homotopy exact sequence associated to this fibration is isomorphic to the exact sequence where [k, −] denotes the Whitehead bracket and T = τ • π * (j) where j is the canonical injection and τ the natural isomorphism

The natural isomorphism
transforms the Whitehead product into the Samelson product, up to a sign, and π * (e) into R : L G (ξ) → Aut G F = G.Then, using corollaries 1 and 2 above, we deduce the exact sequence of groups

Proof of Theorem 3
Let us consider the cochains C * (L * ) on the differential graded Lie algebra L * defined in the introduction, where L ∨ * denotes the graded vector space dual to L By ([20, section 11]), (∧sL ∨ * , d) is a (non minimal) model of B aut G F when G is a maximal subgroup of Aut F acting unipotently on H * (F ; Q).Thus, if F is a nilpotent compactly generated space, Corollary 2 together with ([20, Theorem 10.1]) give the isomorphism If L is a locally nilpotent Lie algebra over Q, we denote by exp(L) the divisible group associated to L by the Campbell-Hausdorff formula Let G be a finitely generated torsion free nilpotent group.In ( [14]), Malcev constructs a Lie algebra L G over the rationals such that G naturally embeds into exp(L G ).The group Ĝ = exp(L G ) is called the Malcev completion of G ( [16], [14]).Let now X be a nilpotent space.The action of π 1 (X) onto π n (X) can be described, modulo the isomorphism π r (X) ∼ = π r−1 (ΩX), by the map Such a space X admits a 0-localization X 0 , which satisfies π 1 (X 0 ) = π 1 (X), π i (X 0 ) = π i (X) ⊗ Q, i ≥ 2.Moreover, the action of π 1 (X) on π * (X) induces an action of the Lie algebra L π1(X) on π n (X) ⊗ Q which is given by the bracket in the Lie algebra π * (ΩX) ⊗ Q ( [2]).
We now return to the particular case, ΩX = aut G F .Let η be a derivation that represents α.We then have exp(H 0 (L * , D) η ) = G α .

Proof of Theorems 4 and 5
The rational homotopy groups π i (L(B, B aut G F , k)) ⊗ Q, i > 1 and the Malcev completion of the nilpotent group π 1 (L(B, B aut G F , k)) can be computed by rational homotopy theory and more precisely by Haefliger's work on mapping spaces ( [10]).In fact, if f : S → T is a continuous map between nilpotent finite type CW complexes, then there exists a complex (D * , ∂), The differential ∂ depends on the map f and the construction is described in ( [10], [8]).
Proof of Theorem 4: When F is an homogeneous space G = K/H, with rank K = rank H, Shiga and Tezuka ( [18]) prove that This implies : and thus ∂ = 0. Therefore, This proves Theorem 4.

Proof of Theorem 5:
We now suppose that Hq (B; Z) is finite for q ≤ n and that π q (F ) is finite for q > n.This implies that In particular, R : L G (ξ) → Ĝ is injective, L 1 (ξ) is a connected space and the evaluation map e : L(B, B aut1 F ; k) → B aut1 F is a rational homotopy equivalence.The commutativity of the following diagram together with Proposition 1 implies now that L 1 (ξ) → aut 1 F is also a rational homotopy equivalence.

L(B, B aut1
Using rational homotopy we can make explicit computations. Proposition 2. Let ξ : E → B be a fibration with fibre F .We suppose that B and F are simply connected finite type CW complexes and that there exists an integer N such that π >N (F ) ⊗ Q = 0, then 1) π n (L 1 (ξ)) is a finite group for n > N.
2) We have isomorphisms Proof: The rational homotopy groups of the space aut 1 (F ) are isomorphic to the homology groups of the space of derivations of the Sullivan minimal model of F ( [20]).It is then clear that π >N (aut 1 (F )) ⊗ Q = 0 and that the evaluation map ev : aut 1 (F ) → F induces an isomorphism on π N (−) ⊗ Q.As B is simply connected, this implies that the vector spaces D n are zero for n > N and for n = N − 1. Therefore we have the isomorphisms π N (L 1 (ξ) ⊗ Q ∼ = D N = H 0 (B; Q) ⊗ π N (aut 1 (F )).
universal fibration for fibrations with fibre F whose "structural group" can be reduced to G.Example.Let B = S n .A Hurewicz fibration ξ = (E, p, B, F ) is determined, up to fibre homotopy, by the homotopy class {α} of a clutching function α : S n−1 → aut F .In this case the structural group of ξ can be reduced to G if and only if for some point p in S n−1 the class [d(p)] belongs to G. Henceforth we shall fix a Hurewicz fibration ξ = (E, p, B, F ) whose base is a CW complex and with classifying map k : B → B aut G F .Because Hurewicz fibrations give rise to a homotopy functor ([1]), and from ([19, Chapitre 7, Section 7, Theorem 11]), we can choose k as an inclusion and ξ as the restriction of (U G ) to B. Let L * (ξ, U G ) be the space of fibre preserving maps from E to B aut • G F which carry each fibre of ξ into a fibre of U G by a homotopy equivalence.Let L * (ξ, U G ; k) be the set of maps in L * (ξ, U) with the additional property that every map f ∈ L * (ξ, U) covers a map B → B aut G F which is homotopic to k.
1. On what conditions is the group L 1 (ξ) finitely generated or finite (rigidity of the fibration) [cf.Theorem 4, below].2. On what conditions is the map R 1 a homotopy equivalence [cf.for instance Theorem 5 below].