On the genus of elliptic fibrations

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 2, Pages 597–606 S 0002-9939(03)07203-4 Article electronically published on August 20, 2003 ON THE GENUS OF ELLIPTIC FIBRATIONS J.-B. GATSINZI (Communicated by Paul Goerss) Abstract. A simply connected topological space is called elliptic if both π∗ (X, Q) and H ∗ (X, Q) are finite-dimensional Q-vector spaces. In this paper, we consider fibrations for which the fibre X is elliptic and H ∗ (X, Q) is evenly graded. We show that in the generic cases, the genus of such a fibration is completely determined by generalized Chern classes of the fibration. Introduction In this paper, all topological spaces are supposed to be 1-connected and having the rational homotopy type of a CW complex of finite type. p The genus of a fibration X → E → B is the least integer n such that B can be covered by n+ 1 open subsets, over each of which p is a trivial fibration, in the sense of fibre homotopy type. We consider here the genus of fibrations whose fibres are elliptic spaces. For recall that a space X is elliptic if both H ∗ (X, Q) and π∗ (X) ⊗ Q are finite-dimensional Q-vector spaces. Through this paper we work over rationals unless otherwise stated, and we will rely on the theory of Sullivan models. We establish the following. p Theorem A. Let X → E → B be a fibration where X is a sphere. Such a fibration is classified by the map f : B → K(Q, 2k). Then the genus of p is the nilpotency index of α = Im H 2k (f ), that is, the least r such that αr+1 = 0 (Theorem 2.3). p Theorem B. Given a fibration X → E → B where X is a homogeneous space G/H, when G and H have the same rank and B is a formal space, the genus of p is bounded above by nil H even (B) (Corollary 4.7). In fact, we prove that the genus of p is equal to the nilpotency index of the subalgebra of H ∗ (B, Q) generated by the generalized Chern classes of the fibration. 1. LS category and related invariants Here we will recall some homotopy invariants of LS category type as well as the relation between the genus and universal fibrations. Received by the editors October 6, 2001 and, in revised form, September 19, 2002. 2000 Mathematics Subject Classification. Primary 55P62; Secondary 55M30. Key words and phrases. Rational homotopy, Lusternik-Schnirelmann category, genus, sectional category. Supported by a grant from Université Catholique de Louvain. c 2003 American Mathematical Society 597 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 598 J.-B. GATSINZI Definition 1.1. The LS category of a space X, written cat(X), is the least integer n such that X can be covered by n + 1 open subsets, each contractible in X. The original definition of the LS category differs from the one above by 1 (see [10]), but the definition above has become a standard in homotopy theory, since cat(X) = 0 if and only if X is contractible. Since a direct computation of cat(X) is difficult, it is more convenient to approximate it by other invariants. Definition 1.2. The nilpotency index of a ring R, denoted by nil(R) is the least integer n such that Rn+1 = 0. If r ∈ R, the nilpotency index of r is the least n such that rn+1 = 0. Note that in our definition, nil(R) is one unit less than the usual definition. We have (1) cat(X) ≥ nil H̃(X), where H̃ is the reduced cohomology with any coefficient ring. Definition 1.3. The category of a map f : X → Y , denoted by cat(f ), is the least integer n such that X can be covered by n + 1 open subsets Ui , for which the restriction of f to each Ui is nullhomotopic. Note that cat(f ) ≤ min{cat(X), cat(Y )}. Moreover, cat(X) = cat(idX ), so that the category of a map is a generalisation of the LS category of a space. As in Equation (1), we have (2) cat(f ) ≥ nil(Im H̃(f )) where H̃(f ) : H̃(Y ) → H̃(X) is the induced morphism in reduced cohomology with any coefficient ring. Definition 1.4. Let p : E → B be a fibration. The sectional category of p, secat(p), is the least integer n such that B can be covered by n + 1 open subsets, over each of which p has a section. An approximation of secat(p) is given by the inequality [10] (3) secat(p) ≥ nil(ker H̃(p)). Definition 1.5. The genus of p is the least integer n such that B can be covered by n + 1 open subsets, over each of which p is a trivial fibration, in the sense of fibre homotopy type. It follows from the definitions above that (4) secat(p) ≤ genus(p), and equality holds if p is a principal fibration. If f : B ′ → B is a map, consider p′ : E ′ → B ′ , the fibration induced from p by f . It is easily seen that secat(p′ ) ≤ secat(p), and equality holds if f is a homotopy equivalence. The genus behaves in a similar way. We define a similar invariant for G-bundles. If p : E → B is a G-bundle, define Gcat(p) as the least integer n such that there is a covering of B by n + 1 open License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE GENUS OF ELLIPTIC FIBRATIONS 599 subsets over each of which p is a trivial bundle. Of course, genus(p) ≤ Gcat(p) and Gcat(p) = genus(π), where π is the associated principal fibre bundle. Moreover, if f : B → BG is the classifying map of π, then [10] (5) Gcat(p) = cat(f ). In view of the relation above, if p is a complex fibre bundle, then Chern classes may play a role in the estimation of the genus(p). We will pursue this analogy for fibrations whose fibres are complex projective spaces. As for Gcat, the genus is closely related to classifying spaces. Recall that fibrations with fibre in the homotopy type of X are obtained, up to fibre homotopy equivalence, as a pull-back of the universal fibration [1] X → B aut• X → B aut X, where aut X denotes the monoid of self-homotopy equivalences of X, aut• X is the monoid of pointed self-homotopy equivalences of X, and B is the Dold-Lashof functor from monoids to topological spaces [2]. Letting B̃ aut X → 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]. The genus behaves like Gcat towards universal fibrations. We have p Theorem 1.6. [10] If X → E → B is a fibration, then (6) genus(p) = cat(f ), where f : B → B aut X is the classifying map of p. Some of the invariants above can also be defined in terms of existence of a section p2 p1 of a fibrewise join of fibrations. If F1 → E1 → B and F2 → E2 → B are fibrations with the same base, then the fibrewise join is the fibration p1 ∗ p2 : E1 ∗B E2 → B, where elements of E1 ∗B E2 are of the form (t1 e1 , t2 e2 ), t1 + t2 = 1, p1 (e1 ) = p2 (e2 ), with the restriction that ti ei is independent of ei if ti = 0. Naturally (p1 ∗ p2 )(t1 e1 , t2 e2 ) = p1 (e1 ) = p2 (e2 ). Note that the fibre is the join F1 ∗ F2 . If p is a fibration, p(n) will denote the fibrewise join of n+1 copies of p. Consider the path fibration γ : P B → B. The total space of the fibrewise join γ(n) will be denoted by Gn (B) and is often referred to as the nth Ganea space of B. The fibration Gn (B) → B is also called the Ganea fibration [7]. Theorem 1.7. Let p : E → B be a fibration and γ : P B → B the path fibration. cat(B) is the least integer n such that γ(n) : Gn (B) → B admits a section [7] and secat(p) is the least integer such that p(n) has a section [10]. In particular, for the path fibration γ : P B → B, cat(B) = secat(γ) = genus(γ). The category of a map can be defined using the Ganea fibration. Theorem 1.8. [7] If f : X → Y is a mapping, cat(f ) is the least n such that there is a lifting f˜ of f in the following diagram: f˜ X f Gn (Y ) < γ(n)  /Y License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 600 J.-B. GATSINZI If one forms the pull-back Enf f¯ / Gn (Y ) γ ′ (n)  X γ(n)  /Y f then cat(f ) is the least n such that the induced fibration Enf → X possesses a section. The rational category of X, denoted by cat0 (X), is defined by cat0 (X) = cat(X0 ). Here X0 denotes the rationalization of X. For a mapping f : X → Y , cat0 (f ) will denote cat(f0 ), where f0 : X0 → Y0 is the rationalization of f . From now on, we will assume that H i (X) is a finite-dimensional Q-vector space, for each i. Recall that the Sullivan minimal model of X is a free commutative cochain algebra (ΛZ, d) such that dZ ⊂ Λ≥2 Z, with Z n ∼ = HomQ (πn (X), Q) (see [16], [9]). Félix and Halperin showed that the rational category can be computed using Sullivan models, by exhibiting a model of the Ganea fibration Gn (X) → X. Theorem 1.9 ([5]). Let f : X → Y be a mapping and f¯ : ΛV → ΛW its Sullivan minimal model. Then cat0 (f ) is the least n such that there is a mapping ρ verifying f¯ = ρ ◦ i in the following diagram. f¯ / ΛW ΛV NN O NNN NNiN ρ p NNN N'  ≃ ΛV /Λ>n V o ΛV ⊗ ΛT In particular, if (ΛZ, d) is the Sullivan minimal model of X, then cat0 (X) is the least integer n such that i has a retraction ρ. (ΛZ, d) j OOO ρ OOO OO p i OOOO  ' ¯ o≃ (ΛZ/Λ>n Z, d) ΛZ ⊗ ΛT Since genus(p) is the category of the classifying map, we recall here the construction of a model of B̃ aut X. If (ΛZ, d) is a Sullivan model of X, then a Lie model of B̃ autX is obtained using derivations on (ΛZ, d). Precisely we define the differential Lie algebra (Der ΛZ, D) as follows [16]: in degree k > 1, take the derivations of ΛZ decreasing the degree by k. In degree one, we only consider the derivations θ that decrease the degree by one and verify ′ dθ + θd = 0. The Lie bracket is defined by [θ, θ′ ] = θθ′ − (−1)|θ||θ | θ′ θ and the differential D is defined by Dθ = [d, θ]. Theorem 1.10 ([16]). The graded differential Lie algebra (Der ΛZ, D) is a Lie model of B̃ aut X. A model of B̃ aut X from the Quillen model of X is found in [13], [18], and [17]. A Sullivan model of the universal fibration is given by the KS extension C ∗ (L) → (C ∗ (L) ⊗ ΛZ, D) → (ΛZ, d) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE GENUS OF ELLIPTIC FIBRATIONS 601 where L = (Der ΛZ, D). The P explicit formula for D is given in [18]. Roughly speaking, for z ∈ Z, Dz = dz + i bi θi (z) where θi are those derivations vanishing on generators of degree greater than |z| and the bi ’s are their duals in C ∗ (L). 2. Spherical fibrations We use Theorem 1.10 to compute a model of B̃ aut X, when X is a sphere. We have the following. Proposition 2.1. If X = S 2n−1 , then (B̃ aut X)0 ≃ K(Q, 2n) and if X = S 2n , then (B̃ aut X)0 ≃ K(Q, 4n). Proof. If X = S 2n−1 , then the Sullivan model of X is (Λx, 0), where |x| = 2n − 1. Hence Der(Λx, 0) = (Q.α, 0) where α is the derivation taking x to 1. Hence B̃ aut X has the rational homotopy type of K(Q, 2n). For X = S 2n , the Sullivan model is (Λ(x, y), d) where |x| = 2n, |y| = 4n − 1, dx = 0, and dy = x2 . If a is a generator of Λ(x, y), let (a, b) denote the derivation of Λ(x, y) taking a to b and vanishing on the other generator. Here the Lie algebra (L, δ) = Der(Λ(x, y), d) is generated (as a vector space) by the derivations α2n−1 = (y, x), α2n = (x, 1), α4n−1 = (y, 1) and the differential is given by δα2n−1 = δα4n−1 = 0, δα2n = 2α2n−1 . Therefore  Hi (L, δ) = Q for i = 4n − 1 and vanishes in all other degrees. If X = S 2n−1 , a straightforward computation shows that a model of the universal fibration X → B̃ aut• X → B̃ aut X is given by the KS extension (Λy2n , 0) → (Λy2n ⊗ Λx2n−1 , d) → (Λx2n−1 , 0) where dy2n = 0, dx2n−1 = y2n . Since (Λy2n ⊗Λx2n−1 , d) is trivial, X → B̃ aut• X → B̃ aut X is rationally equivalent to the path fibration. Therefore every fibration of fibre S 2n−1 is rationally a principal fibration. The sectional category of fibrations of fibre a sphere has been determined by D. Stanley, who proved, among other things, the following. p Theorem 2.2 ([15, Theorem 2.3]). Given a fibration S 2n−1 → E → B with classifying map f : B → K(Q, 2n), if α = Im H 2n (f ), then secat(p) = nil α, that is, the least r such that αr+1 = 0. Since (B̃ aut X)0 ≃ K(Q, 2n) for X = S 2n−1 and (B̃ aut X)0 ≃ K(Q, 4n) for X = S 2n (Proposition 2.1), we can generalize Stanley’s result as follows. p Theorem 2.3. Let X → E → B be a fibration such that B̃ aut X is rationally homotopic to K(Q, 2k). If f : B → K(Q, 2k) is the classifying map of p, then genus(p) = nil α, where α = Im H 2k (f ). Under the hypotheses of Theorem 2.3, suppose that X is an odd sphere. Then the resulting fibration is principal and genus(p) = secat(p) = nil α. We hence recover Theorem 2.2. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 602 J.-B. GATSINZI Proof of Theorem 2.3. The proof of Theorem 2.3 is based on the characterisation of p cat(f ) given by Theorem 1.9. Let X → E → B be a fibration where (B̃ aut X)0 ≃ K(Q, 2k) and (ΛV, d) → (ΛV ⊗ ΛW, D) → (ΛW, D̄) is a KS model of p. The KS extension above is classified by a mapping f : (Λz, 0) → (ΛV, d) with |z| = 2k. Take α = [f (z)] ∈ H 2k (ΛV, d). Suppose that r is the smallest integer such that αr+1 = 0. Since cat(f ) ≥ nil Im H(f ) [10], we conclude that cat(f ) ≥ r. On the other hand, consider the following diagram: f / (ΛV, d) Λz NN O NNN N N NNN p ρ NN&  ≃ (Λ(z, t), d) Λz/(z r+1) o where dt = z r+1 . We define ρ by ρ(z) = f (z) and ρ(t) = β where dβ = (f (z))r+1 . Therefore cat(f ) = genus(p) = r.  One can also prove Theorem 2.2 using the fibrewise join process. If S 2n−1 → p p′ E → B and S 2m−1 → E ′ → X are fibrations, then one can describe a model of  ı p ∗ p′ as follows. Consider the KS extensions B → (B ⊗ Λa, d) and B → (B ⊗ Λb, d) of p and p′ respectively. Note that da is a zero cohomology class in H(B) if and only if p is a trivial fibration. We use the method outlined by Doeraene in [3] to compute a model of the fibre join p ∗ p′ . Consider the push-out B / ı / (B ⊗ Λa, d)   (B ⊗ Λb, d) / ̄  / (B ⊗ Λ(a, b), d) ı̄ ≃ Since ̄ is not surjective, we form (B ⊗ Λa, d) → (B ⊗ Λ(a, c, c̄), d) with dc = c̄ and dc̄ = 0, which is a quasi-isomorphism. We define f : B ⊗ Λ(a, c, c̄) → B ⊗ Λ(a, b) that extends ̄ by setting f (c) = b and f (c̄) = db. Now we form the pull-back ı̃ A f¯  (B ⊗ Λb, d) / / B ⊗ Λ(a, c, c̄) f ı̄  / (B ⊗ Λ(a, b), d) There is a natural mapping B → A that is a commutative model of p ∗ p′ . Recall that A = {(x, y) ∈ (B ⊗ Λb) ⊕ (B ⊗ Λ(a, c, c̄)) | ı̄(x) = f (y)}. Hence a model of the join is the inclusion b 7→ (b, b). Since we know that the homotopic fibre of the fibrewise join is S 2n−1 ∗ S 2m−1 = S 2(m+n)−1 , its model is Λz where |z| = |a| + |b| + 1. If da = α and db = β, then (αβ, αβ) ∈ A is a boundary in A, since d(αb, αc + a(β − c̄)) = (αβ, αβ). Therefore the relative Sullivan model of the fibre join is /A B $JJ O JJ JJ ≃ JJ J$ (B ⊗ Λz, d) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE GENUS OF ELLIPTIC FIBRATIONS 603 with dz = αβ. In particular, p ∗ p′ is a nontrivial fibration if and only if αβ is a nonvanishing cohomology class in H(B). Working by induction, we can then deduce the following. p Proposition 2.4. Let S 2n−1 → E → B be a fibration and B → (B ⊗ Λz, d) its KS-extension, where dz = b ∈ B. Then a model of the n-fibrewise join p(n) = p ∗ · · · ∗ p is given by the KS-extension B → (B ⊗ Λw, d) with dw = bn+1 . In par| {z } (n+1) f actors ticular, secat(p) is the least n such that bn+1 is coboundary in B (see Theorem 2.2). 3. The universal fibration of CP(n) We consider here fibrations with fibre CP(n) of which the Sullivan model is (Λ(a, b), d) with da = 0 and db = an+1 , |a| = 2 and |b| = 2n + 1. To compute the rational homotopy type of B̃ aut X we consider the derivations α2i+1 = (b, an−i ) of Λ(a, b) for i = 0, 1, . . . , n and α2 = (a, 1) (subscripts indicate the degree). As a vector space, the Lie algebra L of derivations of (Λ(a, b), d) is L= n M Qα2i+1 ⊕ Qα2 . i=0 A straightforward computation shows that δα2i+1 = 0 for all 0 ≤ i ≤ n and δα2 = (n + 1)α1 . Hence for 1 ≤ i ≤ n, α2i+1 represents a nonzero homology class in H∗ (L, δ). Therefore n M Qα2i+1 . H∗ (L, δ) = i=1 This implies that the Sullivan minimal model of B̃ aut X is given by (Λ(y4 , y6 , . . . , y2n+2 ), 0) (see also [16, §11]). Note that B̃ aut X has the rational homotopy type of BSU (n + 1). A model for the universal fibration is given by the KS extension (Λ(y4 , y6 , . . . , y2n+2 ), 0) → (Λ(y4 , y6 , . . . , y2n+2 ) ⊗ Λ(a, b), D) → (Λ(a, b), d) with Da = 0, Db = a n+1 + n−1 X ai y2(n+1−i) . i=0 p Let X → E → B be a fibration and (B, d) a Sullivan model of B. The KS model of p, B → (B ⊗ (a, b), D) → (Λ(a, b), d), is classified by a mapping f : (Λ(y4 , y6 , . . . y2n+2 ), 0) → (B, d). Put c4 = [f (y4 )], c6 = [f (y6 )], . . . , c2n+2 = [f (y2n+2 )] ∈ H ∗ (B). We call c4 , c6 , . . . , c2n+2 generalized Chern classes of the fibration p. Denote by r4 , r6 , . . . , r2n+2 their respective nilpotency indexes, that is, rk is the least positive integer such that crkk +1 = 0. We turn to fibrations for which the base is formal. For recall that a space X is formal if there is a morphism (ΛZ, d) → H ∗ (ΛZ, d) that induces an isomorphism in cohomology, where (ΛZ, d) is the Sullivan minimal model of X. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 604 J.-B. GATSINZI p Theorem 3.1. Let CP (n) → E → B be a fibration where B is a formal space. The genus of p is equal to the nilpotency index of Im H(f ), where f is the classifying map of p. In particular, if m = max{ri } and s = r4 + r6 + · · · + r2n+2 , then m ≤ genus(p) ≤ s. Proof. Since B is formal, there is a morphism (ΛZ, d) → H ∗ (B) inducing an isomorphism in homology. Consider the classifying map f : Λ(y4 , y6 , . . . , y2n+2 ) → H ∗ (B). Let k be the nilpotency index of Im f . Since cat(f ) ≥ k by Equation 2, we need only to prove that cat(f ) ≤ k. It is then sufficient to check that f factors through Λ(y4 , y6 , . . . , y2n+2 )/Λ>k (y4 , y6 , . . . , y2n+2 ). If t ∈ Λ>k (y4 , y6 , . . . , y2n+2 ), then t is a finite sum of monomials of the form β 2n+2 , sy4β4 y6β6 . . . y2n+2 s∈Q P βi > k. As a result, there is i such that βi ≥ ri + 1. Therefore where βi ≥ 0 and f (t) = 0 and the result follows.  4. Spaces verifying the Halperin conjecture Definition 4.1. Let X be an elliptic space. The integer X (−1)i dim πi (X) ⊗ Q χπ = i is called the homotopy Euler characteristic of X. Theorem 4.2 ([8]). If X is an elliptic space, then the following statements are equivalent: (1) χπ = 0; (2) H ∗ (X, Q) is concentrated in even degrees. i p Conjecture 4.3 (Halperin). Let X → E → B be a fibration for which X verifies one of the equivalent conditions of Theorem 4.2. Then the (rational) Serre spectral sequence collapses at the E2 level or, equivalently, the morphism H ∗ (i) : H ∗ (E, Q) → H ∗ (X, Q) is surjective. This conjecture has been verified in the following cases: if H ∗ (X, Q) is generated by at most 3 generators [11], [19], if X is a flag manifold [12], and if X is a homogeneous space [14]. In [12] Meier reformulated the conjecture in terms of homotopy groups of classifying spaces. Theorem 4.4. Let X be an elliptic space such that H ∗ (X, Q) is concentrated in even degrees. The following statements are equivalent. i p (1) The Serre spectral sequence for each fibration X → E → B collapses at the E2 level. (2) There is no nonzero negative derivation on the algebra H ∗ (X, Q). (3) If (ΛZ, d) is a Sullivan model of X, then H∗ (Der(ΛZ, d)) is concentrated in odd degrees. (4) π∗ (B̃ aut X) ⊗ Q is concentrated in even degrees. Example 4.5. The space X of which a Sullivan commutative model is of the form Λx1 /(xn1 1 ) ⊗ · · · ⊗ Λxr /(xnr r ), where |xi | is even, satisfies the Halperin conjecture. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE GENUS OF ELLIPTIC FIBRATIONS 605 Consider a fibration of which the fibre is a homogeneous space X = G/H, G and H having the same rank. The Sullivan minimal model of X is of the form (Λ(x1 , . . . , xr , y1 , . . . , yr ), d) where |xi | is even, |yi | is odd and dyi = fi ∈ Λ(x1 , . . . , xr ). In [14], Shiga and Tezuka proved that this space verifies the Halperin conjecture, and hence L = H∗ (Der(Λ(x1 , . . . , xr , y1 , . . . , yr ), d)) is concentrated in odd degrees. Hence the Lie bracket is trivial and B̃ aut X has the rational homotopy type of a product of K(Q, 2k). Take the derivations θ1 , . . . , θn representing homology classes in L. The Sullivan model of B̃ aut X is then given by C ∗ (L) = (Λ(z1 , . . . , zn ), 0), where the zi are of even degree and duals of θi . We denote this model simply by ΛZ. A model of the universal fibration is given by p ΛZ → (ΛZ ⊗ (x1 , . . . , xr , y1 , . . . , yr ), D) → (Λ(x1 , . . . , xr , y1 , . . . , yr ), d) P with Dyi = dyi + j zj θj (yi ) and Dxi = 0 because p is surjective in homology. Let X → E → B be a fibration and (B, d) → (B ⊗ Λ({xi , yi }), D) → (Λ({xi , yi }), d) its KS extension. We have the following push-out, where f is the classifying map of the fibration p. i ΛZ / / (ΛZ ⊗ Λ({xi , yi }), D) / Λ({xi , yi })  / (B ⊗ Λ({xi , yi }), D′ ) / Λ({xi , yi }) f  (B, d) / i′ P Moreover, D′ (yi ) = fi + j f (zj )θj (yi ), where [f (zj )] ∈ H ∗ (B, d). If B is formal, then cat(f ) = nil H(f ), and therefore the genus of p is equal to the nilpotency index of the subalgebra of H ∗ (B) generated by {f (zj )}. Hence we get the following. p Proposition 4.6. Let X → E → B be a rational fibration, where B is a formal space and X has the rational homotopy type of a homogeneous space G/H, G and H having the same rank. The genus of the rationalization of p is equal to the nilpotency ξ∗ index of the subalgebra Im[H ∗ (B̃ aut X) → H ∗ (B)] where ξ : B → B̃ aut X is the classifying map of p. Corollary 4.7. Under the hypotheses of the above proposition, the genus of p : E → B is bounded above by the nilpotency index of H even (B). Remark 4.8. It is not clear how to obtain an upper bound of the sectional category using the index of nilpotency of some cohomology classes. Consider for instance the fibration p of which the KS extension is i Λz2 → (Λz2 ⊗ Λ(x2 , x5 ), d) with dx5 = x32 − z23 . The mapping i admits a retraction r defined by r(x2 ) = r(z2 ) = z2 , r(x5 ) = 0. Hence the sectional category of p is zero, while its genus is infinite by Proposition 3.1. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 606 J.-B. GATSINZI References [1] A. Dold, Halbexakte Homotopiefunktoren, Lecture Notes in Math., no. 12, Springer-Verlag, New York, 1966. MR 33:6622 [2] A. Dold and R. Lashoff, Principal quasi-fibrations and fibre homotopy equivalence of bundles, Illinois J. Math. 3 (1959), 285 − 305. MR 21:331 [3] J.-P. Doeraene, L.S.-category in a model category, J. Pure and Appl. Algebra 84 (1993), 215 − 261. MR 94b:55017 [4] E. Dror and A. Zabrodsky, Unipotency and nilpotency in homotopy equivalences, Topology 18 (1979), 187 − 197. MR 81g:55008 [5] Y. Félix, La dichotomie elliptique-hyperbolique en homotopie rationelle, Astérisque 176, Société Mathématique de France, 1989. MR 91c:55016 [6] Y. Félix and S. 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MR 83c:55016 University of Botswana, Private Bag 0022, Gaborone, Botswana E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use