A note on the Petersen-Wilhelm conjecture
aa r X i v : . [ m a t h . DG ] J un A NOTE ON THE PETERSEN-WILHELM CONJECTURE
DAVID GONZ ´ALEZ- ´ALVARO AND MARCO RADESCHI
Abstract.
In this note we consider submersions from compact manifolds, ho-motopy equivalent to the Eschenburg or Bazaikin spaces of positive curvature.We show that if the submersion is nontrivial, the dimension of the base isgreater than the dimension of the fiber. Together with previous results, thisproves the Petersen-Wilhelm Conjecture for all the known compact manifoldswith positive curvature. Introduction
At the present time there is a very small number of known methods to constructexamples of manifolds with nonnegative or positive (sectional) curvature. More-over, Riemannian submersions are present in the construction of almost all existingexamples of positively curved manifolds (see [23] for a survey on the topic). This isbecause, by the Gray-O’Neill formula, a lower bound for the sectional curvatures ofthe total space also bounds the curvatures of the base space. In the search of newexamples of manifolds with positive curvature, understanding the behavior of Rie-mannian submersions under curvature-related assumptions seems crucial. In thatdirection, the following conjecture has been of great interest in the last decade.
Petersen-Wilhelm’s Conjecture. If M → B is a Riemannian submersion be-tween compact, positively curved manifolds, with fiber F , then dim F < dim B . It is worth mentioning that, although this conjecture has been attributed inthe literature to Fred Wilhelm, during the preparation of this article the authorslearned that it was originally considered jointly with Peter Petersen.The conjecture is known to hold under the extra assumption that at least twoof the fibers are totally geodesic, thanks to Frankel’s theorem. On the other hand,there exist counterexamples if one weakens the conjecture to require positive sec-tional curvature on an open and dense set of the total space (cf. [17] and [5, p.167]). We refer the reader to [14, 20] for recent progress towards the conjecture inthe general case, and to [12] for a complete list of all the (few) currently knownRiemannian submersions between positively curved manifolds.The goal of this note is to show that every Riemannian submersion from anyof the known compact positively curved manifolds satisfies the Petersen-Wilhelmconjecture. In fact, the results here will show that any submersion (not necessarilyRiemannian) from any compact manifold homotopy equivalent to one of the knownpositively curved examples, satisfies the conjecture.
Mathematics Subject Classification.
Recall that, by Bonnet-Myers Theorem, positively curved manifolds have finitefundamental group and therefore for our purposes it is enough to study simplyconnected examples. For the convenience of the reader, we briefly review all existingexamples of simply connected manifolds with positive curvature. • Spaces admitting a homogeneous metric of positive curvature. These havebeen classified (see [22]) and include the classical compact rank one sym-metric spaces, three single examples due to Wallach, two single examplesdue to Berger and an infinite family due to Aloff-Wallach: S n , CP n , HP n , Ca P , W , W , W , B , B , W p,q . • Eschenburg [8] constructed positively curved metrics on a biquotient space E of dimension 6; and on an infinite family of biquotients of dimension 7,which are circle quotients SU(3) //S and include the Aloff-Wallach spaces W p,q as a subfamily. • Bazaikin [3] constructed positively curved metrics on an infinite family ofbiquotients SU(5) // Sp(2) S of dimension 13, which contains the Bergerspace B as a particular case. • Dearricot [6] and Grove-Verdiani-Ziller [15] recently constructed a cohomo-geneity one positively curved manifold of dimension 7, denoted P .In a recent work, Amann and Kennard studied a generalized version of thePetersen-Wilhelm conjecture. Given a fibration F → M → B , they gave topologicalconditions for M to ensure that dim F < dim B (see Theorem A in [2]). Theseconditions are satisfied by any manifold that is rationally homotopy equivalentto any of the simply connected compact rank one symmetric spaces or to any of W , W , W . Recall that E is rationally homotopy equivalent to W , and that B and P are rational spheres. It follows that all of these examples satisfy (astronger version of) the Petersen-Wilhelm conjecture. In this note we deal with theremaining examples, namely the Eschenburg biquotients of dimension 7, and theBazaikin biquotients, for which we prove the following results. Theorem A.
Let M be a compact manifold, homotopy equivalent to a -dimensionalEschenburg space. Then there are no submersions from M to a manifold of di-mension ≤ . Theorem B.
Let B be a compact manifold, homotopy equivalent to a Bazaikinspace. Then there are no submersions from B to a manifold of dimension ≤ . It follows that, were to exist a counterexample to the Petersen-Wilhelm con-jecture, it would need to come from a new positively curved manifold, not evenhomotopy equivalent to any of the known ones. This shows the complexity ofthe problem, since finding new examples has been, and continues to be, a highlychallenging task.The theorems above, together with the results in [2], imply the following.
Corollary C.
Any nontrivial submersion from a compact manifold, homotopyequivalent to any of the known compact positively curved manifolds, satisfies thePetersen-Wilhelm’s conjecture.
Observe that the bounds in Theorems A and B are optimal, in the sense thatthe Eschenburg and the Bazaikin spaces admit submersions to CP and CP re-spectively. NOTE ON THE PETERSEN-WILHELM CONJECTURE 3
The proofs of Theorems A and B are remarkably similar, and follow from moregeneral results. Recall that the Eschenburg and the Bazaikin spaces split rationallyas S × S and CP × S respectively (cf. [4]). For each of these spaces, we applytools from rational homotopy theory to rule out all potential submersions exceptthose over S and CP respectively. Ruling out these last possibilities essentiallyconstitutes the mathematical core of this note.The paper is organized as follows. In Section 2 we recall the main results fromrational homotopy theory that will be used later on. Then in Sections 3 and 4 weprove Theorems A and B, respectively. Acknowledgements.
This project grew out of a conversation with Karsten Grove,while the first author was visiting the University of Notre Dame. The first authorwishes to thank the University of Notre Dame for the hospitality during his stay.2.
Basics on rational homotopy theory
In this section we recall the basic facts about rational homotopy theory, that willbe used later on. For more details on the theory, the reader is referred to [10] and[1]. All algebras and vector spaces in this section are over the field Q of rationalnumbers.2.1. Differential graded algebras.
Recall that a differential graded algebra (alsocalled dga ) consists of a graded commutative algebra A = ⊕ i A i , together with adegree-1 map (a differential ) d : A → A that satisfies the Leibniz rule, and d ◦ d = 0.A morphism of dga’s ϕ : ( A, d ) → ( B, d ) is a graded algebra morphism ϕ : A → B ,commuting with the differentials.Since d ◦ d = 0, it is possible to define the cohomology of a dga H ∗ ( A, d ) As thehomology of the complex (
A, d ), and one defines the cohomological dimension of(
A, d ) to be the largest integer n (possibly ∞ ) such that H n ( A, d ) = 0.2.2. Sullivan functor.
The
Sullivan functor A P L is a contravariant functor thatassociates, to any path-connected pointed space X (resp. to any map f : X → Y between path connected pointed spaces) a dga ( A P L ( X ) , d ) given by the rationalPL-forms on the singular simplices of X (resp. a morphism f ∗ : ( A P L ( Y ) , d ) → ( A P L ( X ) , d ) of dga’s obtained by pull-back of PL-forms). The Sullivan functor hasthe property that, for any path connected pointed space X , one has H ∗ ( A P L ( X ) , d ) ≃ H ∗ ( X, Q ) . Sullivan algebras.
Given a Z -graded vector space V , let V even (resp. V odd )denote the subspace of V spanned by the elements of even (resp. odd) degree, anddefine the free commutative graded algebra Λ V byΛ V = ∧ V odd ⊗ Q [ V even ] , where ∧ V odd denotes exterior algebra and Q [ V even ] denotes polynomial algebra. A Sullivan algebra is a dga of the form (Λ
V, d ), satisfying d (Λ V ) ⊆ Λ + V · Λ + V where Λ + V ⊂ Λ V denotes the ideal generated by the elements of strictly positivedegree. D. GONZ´ALEZ-´ALVARO AND M. RADESCHI
Nilpotent spaces, and minimal model.
Let now X be a CW-complex.Recall that X is called nilpotent if π ( X ) is nilpotent, and the action of π ( X ) on π i ( X ) is nilpotent for all i ≥
1. For a nilpotent CW-complex X , there exists a space X Q , called the rationalization of X , such that H ∗ ( X Q , Z ) ≃ H ∗ ( X ; Q ), togetherwith a map X → X Q inducing isomorphisms π k ( X ) ⊗ Q → π k ( X Q ) ⊗ Q . Twonilpotent CW-complexes are rationally homotopy equivalent , if the correspondingrationalizations are homotopy equivalent. Rationally homotopy equivalent spaceshave, in particular, isomorphic rational homotopy groups π Q i ( X ) := π i ( X ) ⊗ Q andisomorphic rational cohomology rings.Given a nilpotent CW-complex, one can also define a Sullivan algebra (Λ V X , d )and a morphism ϕ : (Λ V X , d ) → ( A P L ( X ) , d ) of dga’s, such that the induced mapin cohomology ϕ ∗ : H ∗ (Λ V X , d ) → H ∗ ( A P L ( X ) , d ) ≃ H ∗ ( X, Q ) is an isomorphism.The algebra (Λ V X , d ) is called Sullivan model (also minimal model ) of X , and itsatisfies the following remarkable properties: • The minimal model is unique up to isomorphism. • If π ( X ) is abelian, there are isomorphisms V iX ≃ π Q i ( X ) for all i ≥ V iX ⊂ V X is the subspace of degree i , and π Q i ( X ) denotes π i ( X ) ⊗ Q (cf. [1, Theorem (2.3.7)]). • The minimal model determines X up to rational homotopy equivalence: if Y is another nilpotent CW-complex, its minimal model is isomorphic to(Λ V X , d ) if and only if X, Y are rationally homotopy equivalent.2.5.
Minimal models and fibrations.
Minimal models behave nicely with re-spect to fibrations. In particular, given a fibration F → M → X of nilpo-tent spaces, where F and X have minimal models (Λ V F , d F ) and (Λ V X , d X ) re-spectively, then there exists a dga (Λ V F ⊗ Λ V X , D ) and a morphisms of dga’s(Λ V F ⊗ Λ V X , D ) → ( A P L ( M ) , d ) inducing isomorphism in cohomology (see [10,Sec. 15(a)]). This model comes together with dga maps(Λ V X , d X ) ⊗ id −→ (Λ V F ⊗ Λ V X , D ) id ⊗ ǫ −→ (Λ V F , d F ) , where ǫ : Λ V X → Q sends Λ + V X to 0, which induce in cohomology the expectedmaps H ∗ ( X, Q ) → H ∗ ( M, Q ) → H ∗ ( F, Q ). The dga (Λ V F ⊗ Λ V X , D ) is not neces-sarily a minimal model, but it is called relative minimal model of M .2.6. Rationally elliptic spaces.
Let X be a nilpotent CW-complex, with π ( X )abelian. Then X is called rationally elliptic , if ∞ X i =0 dim H i ( X, Q ) < ∞ , ∞ X i =1 dim π Q i ( X ) ≤ ∞ . Friedlander and Halperin proved in [13] that rationally elliptic spaces have veryrestrictive rational homotopy groups. In particular, the list of possible rationalgroups of rationally elliptic spaces of dimension ≤
7, computed in [19] (for dimension ≤
6) and [7] (in dimension 7) is exceptionally limited, and it is shown in Table 1 atthe end of this paper.All the known compact simply connected manifolds with non-negative curvatureare rationally elliptic (and in fact, the so-called Bott-Grove-Halperin conjecturestates that this should always be the case). In particular, we will be dealing withsubmersions π : M → X where M is a compact, simply connected, rationallyelliptic manifold. NOTE ON THE PETERSEN-WILHELM CONJECTURE 5
The following result is well known, and it will be crucial to rule out most sub-mersions from Eschenburg and Bazaikin biquotients.
Proposition 1.
Suppose M , X are compact simply connected manifolds, and π : M → X is a submersion with fiber F . Then:a. F is a compact nilpotent space, with abelian fundamental group.b. If M is rationally elliptic, so are X and F , and the dimension of F can becomputed from the ranks f i = dim π Q i ( F ) , by dim F = ∞ X i =1 (2 i + 1) f i +1 − ∞ X i =1 (2 i − f i . (1) Proof. a. Since π : M → X is a submersion between compact manifolds, it is alsoa locally trivial fiber bundle (in particular a fibration). The generic fiber, denotedby F , is a compact manifold and, being the fiber of a fibration from a nilpotentspace, it is itself nilpotent (cf. [16, Theorem 2.2]). Moreover, from the long exactsequence in homotopy for the fibration F → M → X it follows that π ( F ) is abelian.b. It follows from the previous point that the spaces M , X , F admit minimalmodels (Λ V M , d ), (Λ V X , d ), (Λ V F , d ) respectively. Since M , X are simply con-nected and π ( F ) is abelian, it follows from Section 2.4 that V iM , (resp. V iX , V iF ) isisomorphic to π Q i ( M ) (resp. π Q i ( X ), π Q i ( F )) for all i ≥ M is rationally elliptic if and only if the rational Betti numbers b i (Ω M )of the loop space of M grow at most polynomially in i . The E -page of the Leray-Serre spectral sequence for the fibration Ω M → Ω X → F is given by E p,q = H p (Ω M, Q ) ⊗ H q ( F, Q ), and thereforedim H i (Ω X, Q ) ≤ X p − q = i dim H p (Ω M, Q ) dim H q ( F, Q ) ≤ C dim F X k =0 b i − k (Ω M )where C = max j b j ( F ). Because M is rationally elliptic, by Proposition 33.9 in [10]it follows that dim F X k =0 b i − k (Ω M ) ≤ Bi m for some constant B and some positive integer m . Therefore, b i (Ω X ) ≤ BC i m , and P ki =1 b i (Ω X ) ≤ BC k m +1 , so X is rationally elliptic. In particular, P i dim( π Q i ( X ))is finite. From the long exact sequence in rational homotopy, the same holds for F and thus F is rationally elliptic.Finally, since F is rationally elliptic, Equation (2.2) in [13] gives that the largestinteger n F such that H n F (Λ V F , d ) = H n F ( F, Q ) = 0 can be computed from the f i = dim V iF = dim π Q i ( F ), as n F = ∞ X i =1 (2 i + 1) f i +1 − ∞ X i =1 (2 i − f i . Since F is compact and orientable, n F = dim F and Equation (1) follows. (cid:3) D. GONZ´ALEZ-´ALVARO AND M. RADESCHI Submersions from 7-dimensional examples
The goal of this section is to prove Theorem A. This result will follow from themore topological Propositions 2 and 3 below.
Proposition 2.
Let M be a compact, simply connected manifold that is rationallyhomotopy equivalent to S × S . If π : M → X is a submersion onto a simplyconnected compact manifold, then X has the rational homotopy groups of either S , CP , S , or S × CP .Proof. Since M is rationally homotopy equivalent to S × S , in particular it isrationally elliptic, and by Proposition 1, the same holds for X and F . Because X is simply connected, rationally elliptic and has dim X ≤
6, there is a finite list ofpossibilities for the ranks c i = dim π Q i ( X ) of the rational homotopy groups of X .We list these possibilities in Table 1, together with the corresponding dimension n of X , computed using Equation (1).Suppose that the rational homotopy groups of X are not the ones of S , CP , S or S × CP . We will show by contradiction that such a space cannot occur.For each of these possible X , one can easily compute the corresponding rationalhomotopy groups π Q i ( F ) = Q f i of F via the long exact sequence in (rational)homotopy for the fibration F → M → X . For instance, suppose that X has therational homotopy groups of S . In this case the fiber F is 4-dimensional, and thenon-zero ranks f i = dim π Q i ( F ) are either:(1) f = f = 1, or(2) f = f = 1, f = 2.In the first case, the models for X and F would be of the type (Λ( x ) , d ) and(Λ( y , y ) , d ) respectively, and thus a relative model for M (cf. Section 2.5) wouldbe (Λ( y , x , y ) , D ) with D ( x ) = 0. In this case however, y would representa nonzero element of H ( M, Q ), in contradiction with the fact that H ( M, Q ) = H ( S × S , Q ) = 0.In the second case, using Equation (1), one obtains that the cohomological di-mension of H ∗ (Λ V F , d ) ≃ H ∗ ( F, Q ) would be 6, in contradiction with the fact thatthe cohomological dimension of compact orientable manifolds agrees with the usualdimension.In all remaining cases for the rational homotopy groups of X , one obtains acontradiction as in the second case in the example above. Namely, one computesall possible f i = dim π Q i ( F ), and uses Equation (1) to compute the cohomologicaldimension of H ∗ (Λ V F , d ) ≃ H ∗ ( F, Q ), which never agrees with the actual dimensionof F . (cid:3) Notice that all the possibilities listed above can occur, as quotients of S × S . Proposition 3.
Let M be as in Proposition 2 and assume in addition that π ( M ) = Z . Then M does not admit submersions onto any manifold of dimension ≤ .Proof. Since M is simply connected, we can restrict our attention to submersionsonly simply connected manifolds. By Proposition 2, the only possible sumbersionfrom M is onto S .Suppose that there is a submersion π : M → S , with fiber F . Since M isrationally homotopy equivalent to S × S , it follows from the long exact sequence NOTE ON THE PETERSEN-WILHELM CONJECTURE 7 in rational homotopy that the nonzero ranks f i of the rational homotopy groups π Q i ( F ) = Q f i have to be one of the following:(1) f = 1, or(2) f = f = f = 1, or(3) f = f = f = 1.The latter cannot occur, since by (1), the cohomological dimension of F equals7 = 5.To rule out the remaining cases, we need to consider the Leray-Serre spectralsequence in cohomology of the fibration M → S , with coefficients in R = Q or Z .Since the base of the fibration is S , the elements in the E -page are nonzero onlyin the 0-th and 2-nd columns, and we obtain a long exact sequence of the form0 → H ( M, R ) → H ( F, R ) → H ( F, R )(2) β → H ( M, R ) → H ( F, R ) → H ( F, R ) → H ( M, R ) → . . . For the case f = f = f = 1, the fact that π Q ( F ) = Q implies that H ( F, Q ) = Q , and we obtain the following contradiction. On the one hand, since H ( M, Q ) = H ( F, Q ) = Q and H ( M, Q ) = H ( M, Q ) = 0, it follows from the long exactsequence (2) with rational coefficients that H ( F, Q ) = Q . On the other hand, theminimal model (Λ V F , d ) of F equals (Λ( x , x , x ) , d ) for some differential d , andin particular dim H ( F, Q ) = dim H (Λ V F , d ) ≤ dim Λ V F = 1.The rest of the proof is dedicated to rule out the possibility f = 1. In thiscase F has the same model as S , and the groups H i ( F, Z ) are finite for i =1 , . . . ,
4. It follows from the long exact sequence (2) with integer coefficients that themap β : H ( F, Z ) → H ( M, Z ) is nonzero. Observe that under the identification H ( F, Z ) ∼ = H ( S , Z ) from the E -page of the spectral sequence, the map β isequivalent to π ∗ : H ( S , Z ) = Z → H ( M, Z ) = Z . Therefore, the generator g of H ( S , Z ) is sent to k ¯ g , where ¯ g is a generator of H ( M, Z ) and k is a positive integer.Recall that principal S -bundles over a manifold X are in bijective correspon-dence with the elements of H ( X, Z ), via the first Chern class. The generator g corresponds to the Hopf fibration S → S . Moreover, letting P → M denote theprincipal S -bundle with Chern class ¯ g , it is easily checked from the Gysin sequenceof S → P → M that P is 2-connected. By taking a cyclic subgroup Z k ⊂ S ,the quotient P/ Z k is the total space of an S bundle P/ Z k → M , with first Chernclass k ¯ g . Therefore, the bundle π ∗ S → M is isomorphic to P/ Z k → M , and inparticular π ∗ S is homeomorphic to P/ Z k .The submersion ˆ π : π ∗ S → S , whose fibers are diffeomorphic to F , lifts toa submersion ˜ π : P ′ → S where P ′ denotes the universal cover of π ∗ S . Here P ′ is homeomorphic to P , is 2-connected, and the fibers are diffeomorphic to theuniversal cover ˜ F of F . Observe that ˜ F is a simply connected rational sphere.Let j : ˜ F → P ′ be the inclusion of a fiber of ˜ π . Since P ′ → S is a submersion, onehas the well-known equality j ∗ ( w ( P ′ )) = w ( ˜ F ) on the total Stiefel-Whitney classes.However, since P ′ is 2-connected, one has w ( P ′ ) = 0 and therefore w ( ˜ F ) = 0. Inother words, ˜ F is a simply connected, compact, spin 5-manifold. D. GONZ´ALEZ-´ALVARO AND M. RADESCHI
It follows from Smale’s classification of simply connected, spin 5-manifolds thatthe torsion of H ( ˜ F , Z ) = H ( ˜ F , Z ) is isomorphic to a sum L i Z k i ⊕ Z k i . The longexact sequence (2) in integer cohomology for ˜ π : P ′ → S yields · · · → H ( ˜ F , Z ) → H ( ˜ F , Z ) → H ( P ′ , Z ) → . . . Since H ( P ′ , Z ) = H ( P ′ , Z ) = 0 and H ( ˜ F , Z ) = Z , it follows that H ( ˜ F , Z ) isfinite and cyclic, thus in order not to contradict Smale’s result, one must have H ( ˜ F , Z ) = 0, and this implies that ˜ F is diffeomorphic to the sphere S . However,from the long exact sequence for ˜ F → P ′ → S we obtain π ( P ′ ) ≃ π ( S ) = Z .From the long exact sequence of S → P ′ → M it follows that π ( P ′ ) = π ( M )and thus π ( M ) ≃ Z , in contradiction with the assumption. (cid:3) Proof of Theorem A.
Let M be homotopy equivalent to an Eschenburg space M ′ .By Belegradek and Kapovitch (cf. [4, Lemma 8.2]), M ′ (and thus M ) is rationallyhomotopy equivalent to S × S , and by Proposition 31 in [8], π ( M ) = π ( M ′ ) = 0.The result now follows from Proposition 3.4. Submersions from 13-dimensional examples
The goal of this section is to prove Theorem B. This result will follow from themore topological Propositions 4 and 5 below.
Proposition 4.
Let B be a compact, simply connected manifold that is rationallyhomotopy equivalent to S × CP . If π : B → X is a submersion onto a simplyconnected compact manifold X of dimension ≤ , then X has the rational homotopygroups of CP .Proof. This proposition is proved along the same lines as the proof of Proposition 2:since B is rationally homotopy equivalent to S × CP , in particular it is rationallyelliptic and by Proposition 1 so are X and F . Since X is compact, simply connectedand with dim X ≤
7, the rational homotopy groups of X fall into the finite list inTable 1. Assume moreover that X does not have the rational homotopy groupsof CP . For any such case, we use the long exact sequence in rational homotopyfor the submersion B → X to compute the ranks f i = dim π Q i ( F ) of the fiber F .Finally, we use Equation (1) to compute the cohomological dimension of H ∗ ( F, Q ),which never agrees with the dimension of F with the exception of one case, namelyif X has the rational homotopy groups of S , and f = f = 1.In this case, the minimal models of X and F are respectively(Λ V X , d ) = (Λ( x ) , d = 0) , (Λ V F , d ) = (Λ( z , z ) , dz = 0 , dz = z ) . Then a relative model for B is (Λ V B , D ) = (Λ( z , x , z ) , D ), where D ( x ) = 0. Inthis case, since D (Λ V B ) = 0, x represents a nonzero element in H (Λ V B , D ) = H ( B, Q ), contradicting the fact that H ( B, Q ) = H ( S × CP , Q ) = 0. (cid:3) Observe that for B as in Proposition 4, we have that H ( B, Z ) = 0 and H ( B, Z ) = Z . Proposition 5.
Let B be as in Proposition 4, and assume in addition that thetruncated cohomology ring H ≤ ( B, Z ) is isomorphic to the integral cohomology of NOTE ON THE PETERSEN-WILHELM CONJECTURE 9 CP . Then if B admits a submersion onto a manifold X of dimension ≤ , it mustbe X ≃ CP and π i ( B ) ≃ π i ( S ) for i = 3 , . . . .Proof. Suppose there is a submersion B → X where X has dimension at most 7,where as usual we can assume that X is simply connected. Then by Proposition 4the base space X has the rational homotopy groups of CP , and by Lemma 3.2 of[18], X is homeomorphic to CP .From the long exact sequence of the fibration B → X , the ranks f i of the rationalhomotopy groups of the fiber F must be either:(1) f = f = f = 1, or(2) f = 1.In the first case, the minimal models of X and F are, respectively,(Λ( x , x ) , dx = 0 , dx = x ) , (Λ( z , z , z ) , dz = dz = 0 , dz = z ) . A relative model for B is then(Λ V B , D ) = (Λ( z , z , x , x , z ) , D ) , with D ( x ) = 0, D ( x ) = x and D ( z ) = ax with a ∈ { , } (cf. Section 2.5).Since H ( B, Q ) = 0, it must be D ( z ) = x . Then, since H ( B, Q ) = Q , it must be D ( z ) = 0. However, in this way D ( z ) = 0, but Λ V B = span( x , z z , x z , z x z )and D (Λ V B ) = span( x , x z , x z ). In particular, z does not lie in D (Λ V B ),and therefore it defines a nonzero class in H (Λ V B , D ) ≃ H ( B, Q ), in contradictionwith the fact that H ( B, Q ) = 0.For the rest of the proof we assume to be in the second case, and the argumentgoes along the same lines as the proof of Proposition 3. Notice that F is, in thiscase, a rational sphere. Using the Leray-Serre spectral sequence of the fibration π : B → X , we deduce that the map π ∗ : H ( X, Z ) → H ( B, Z ) is nonzero. Agenerator of H ( X, Z ) ≃ Z corresponds to the first Chern class of the Hopf fibration S → CP ≃ X . Now, let E → B denote the circle bundle whose first Chern classis a generator of H ( B, Z ) ≃ Z , and observe that in particular π ( E ) = 0. Then π can be lifted, up to homotopy, to a fibration p : E → S , and whose fiber ˜ F is theuniversal cover of F and hence a simply connected rational sphere.We can obtain some information about the topology of E from the properties ofthe Gysin sequence of S → E → B . Since the first Chern class of a circle bundleequals its Euler class, it follows from the assumptions on the cohomology of B that H i ( E, Z ) = 0 for i ≤ H ( E, Z ) = Z . In particular, E is 4-connected.Next we study the topology of the fiber ˜ F . Consider the Leray-Serre spectralsequence in integral cohomology of the fibration ˜ F → E p → S . The elements inthe E -page are nonzero only in the 0-th and 5-th columns, thus we obtain a longexact sequence0 → H ( E, Z ) → H ( ˜ F , Z ) → H ( ˜ F , Z )(3) → H ( E, Z ) → H ( ˜ F , Z ) → H ( ˜ F , Z ) → H ( E, Z ) → . . . It is straightforward to compute the cohomology of ˜ F from (3): H i ( ˜ F , Z ) = Z , i = 0 , Z r , i = 5; for some integer r ≥ , , otherwise. By Poincar´e Duality, ˜ F is 3-connected and H ( ˜ F , Z ) = H ( ˜ F , Z ) = Z r .We claim that Z r = 0. In fact, notice first that ˜ F is a ( s − s + 1)-manifolds, with s = 4. By the seminal work by Wall on highly connected manifolds[21, Cor. 2], there exists a nondegenerate bilinear form b : H ( ˜ F , Z ) × H ( ˜ F , Z ) → Q / Z . which is, in this case, strongly skew-symmetric, which means that b ( x, x ) = 0 forevery x ∈ H ( ˜ F , Z ). Since in this case H ( ˜ F , Z ) ≃ Z r is cyclic, it must be b = 0: infact, for any [ m ] , [ n ] ∈ Z r , we have b ([ m ] , [ n ]) = mn · b ([1] , [1]) = 0. However, since b is nondegenerate, it follows that Z r = 0, as claimed.By Poincar´e Duality, Hurewicz Theorem, and Whitehead Theorem, it then fol-lows that ˜ F is homotopy equivalent (hence homeomorphic) to a sphere S and,from the long exact sequence in homotopy for ˜ F → E → S , we obtain that π i ( E ) → π i ( S ) is an isomorphism for i = 1 . . .
8. On the other hand, it followsfrom the long sequence in homotopy for S → E → B that π i ( B ) ≃ π i ( E ) for i = 3 , . . .
8, and the result follows. (cid:3)
Proof of Theorem B.
Let B be homotopy equivalent to a Bazaikin space B ′ .By Belegradek and Kapovitch (cf. [4, Lemma 8.2]), B ′ (and thus B ) is rationallyhomotopy equivalent to CP × S . Moreover, one can check in [11] that the Bazaikinspaces satisfy the conditions on the cohomology ring required in Proposition 5, but π ( B ′ ) = 0 = Z = π ( S ). The result now follows from Proposition 5. n Example for X n c i = dim π Q i ( X )2 S c = c = 13 S c = 14 S c = c = 1 CP c = c = 1 S × S or CP ♯ CP c = c = 25 S c = 1 S × S c = 1 , c = 26 CP c = c = 1 S × S c = 2 S c = c = 1 S × S c = c = c = c = 1 S × S × S c = c = 3 W or S × CP c = c = 1 , c = 27 S c = 1 S × S c = c = c = 1 S × S c = c = c = 1 S × S × S c = 2 , c = 3 Table 1.
Possibilities for the rational homotopy groups of a lowdimensional space.
NOTE ON THE PETERSEN-WILHELM CONJECTURE 11
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