HHOMOLOGY VERSUS HOMOTOPY IN FIBRATIONSAND IN LIMITS
MANUEL AMANN
Abstract.
Motivated by prominent problems like the Hilali conjectureYamaguchi–Yokura recently proposed certain estimates on the relationsof the dimensions of rational homotopy and rational cohomology groupsof fibre, base and total spaces in a fibration of rationally elliptic spaces.In this article we prove these estimates in the category of formal ellipticspaces and, in general, whenever the total space in addition has positiveEuler characteristic or has the rational homotopy type of a homogeneousmanifold (respectively of a known example) of positive sectional curvature.Additionally, we provide general estimates approximating the conjecturedones.Moreover, we suggest to study families of rationally elliptic spacesunder certain asymptotics, and we discuss the conjectured estimatesfrom this perspective for two-stage spaces.
Introduction
The Hilali conjecture ([8]) speculates that for a rationally elliptic space,i.e. a simply-connected space with both finite rational cohomology andrational homotopy groups, the dimension of rational cohomology is at leastas large as the dimension of the rational homotopy groups, i.e. in other words,their well-defined quotient h ( X ) = dim π ∗ ( X ) ⊗ Q dim H ∗ ( X )is well-defined and smaller equal one. While the conjecture still being open,this quotient was considered in several different further circumstances (forexample see [16]).Recently, it was asked by Yamaguchi–Yokura how this quotient behavesin fibrations of rationally elliptic spaces.It is the goal of this article to provide several special cases of their con-jectured estimates on the one hand and, on the other hand, to study thisquotient asymptotically—first suggesting, specifying and discussing differentreasonable notions of “asymptotic behaviour” for families of rationally ellipticspaces.Throughout this article we denote by X a simply-connected CW-complex.Cohomology is considered with rational coefficients. As stated above, we call Date : June 1st, 2020.2010
Mathematics Subject Classification.
Key words and phrases. fibrations, Hilali conjecture, rational homotopy, rational co-homology, elliptic spaces, formal elliptic spaces, asymptotic behaviour, positively curvedmanifolds. a r X i v : . [ m a t h . A T ] J un MANUEL AMANN X rationally elliptic if it is simply-connected and both dim π ∗ ( X ) ⊗ Q < ∞ and dim H ∗ ( X ) < ∞ . It is called F or positively elliptic , if χ ( X ) >
0. (Byabuse of notation, we shall refer to rationally elliptic spaces as just being elliptic . This is also indebted to the fact that the article builds on rationalmethods. In particular, whenever we speak of a “fibration”, it is actuallyenough to have a “rational fibration” structure.)Recall that the prominent subclass of elliptic spaces, the class of two-stage spaces , is defined as follows: Their minimal Sullivan models (Λ V, d)up to isomorphism admit decompositions of the form V odd = W ⊕ W andd( V even ⊕ W ) = 0, d W ⊆ Λ( V even ⊕ W ).We provide several notions of convergence for families of elliptic spaces.See Section 2 for an elaborate discussion of this. In particular, we rigorouslydefine π -convergence there. It appears to be very hard to control the possiblevalues of h ( X ); so it seems reasonable to consider its asymptotic behaviour.To our knowledge this is the first time such a discussion is launched. As afirst step this then permits to prove Theorem A.
The family of two-stage spaces X π -converges to , i.e. with dim π ∗ ( X ) ⊗ Q (for X ∈ X ) tending to ∞ , h ( X ) tends to . The following results deal with the behaviour of h ( X ) in fibrations. Henceconsider a fibration F (cid:44) → X → B of rationally elliptic spaces. Conjecture 0.1 (Yamaguchi–Yokura, [22]) . · h ( F × B ) ≤ h ( X ) < h ( F ) + h ( B ) + 14(1)As an application of Theorem A we discuss an asymptotic version of thisproblem first. We say that a class X of rationally elliptic spaces asymptoticallysatisfies Conjecture 0.1 if the following holds: There is k ∈ N such that ifdim π ∗ ( X ) ⊗ Q ≥ k for X ∈ X , then X satisfies the conjecture.Clearly, any family X π -converging to 0 asymptotically satisfies the righthand side equation, i.e. for X ∈ X with large enough rational homotopygroups, the quotient h ( X ) is smaller than 1 / Corollary B.
Let X be a family π -convergent to . Then X asymptoticallysatisfies the right hand side of Inequality (1) . In particular, this holds truefor two-stage spaces. Clearly, there are two-stage spaces (already products of spheres) of ar-bitrarily large rational homotopy. Actually, our estimates for two-stagespaces are explicit, and it is easy from there to provide concrete numbers fordim π ∗ ( X ) ⊗ Q starting the range in which the inequality holds.The following results aim to verify Conjecture 0.1 in particular cases. Asa vital tool to do so we first verify a more or less close approximation to theleft hand side of Conjecture 0.1 in Theorem C.
For any fibration
F (cid:44) → X → B of elliptic spaces it holds that h ( F × B ) ≤ · h ( X )Next we prove the conjecture whenever X is positively elliptic, as well asin the category of formal elliptic spaces. Recall that a space is formal if its OMOLOGY VS. HOMOTOPY 3 rational homotopy type is isomorphic to the one of its cohomology algebra.Formal spaces form one of the most prominent classes of spaces in rationalhomotopy theory.
Theorem D.
Let
F (cid:44) → X → B be a fibration of elliptic spaces.If F is an F -space it holds that h ( F × B ) ≤ · h ( X ) Moreover, Conjecture 0.1 holds with respect to any such fibration whenever • X is an F -space, or • F is an F -space and satisfies the Halperin conjecture. For a brief discussion of the Halperin conjecture see Section 1.1. Inparticular, there we provide a list of several classes of spaces for which theconjecture is verified.The last formulation is stricter than necessary: For a fixed totally non-homologous to zero fibration
F (cid:44) → X → B with F an F -space the requiredinequalities hold already. With the presented formulation it is our goal tostress that conjecturally any F -space F should render the fibration totallynon-homologous to zero.Combining, refining and extending the previous arguments we finallyobtain Theorem E.
Conjecture 0.1 holds for a fibration
F (cid:44) → X → B of ellipticformal spaces. This is proved in Propositions 5.3 and 5.4.As a corollary to this we can prove Conjecture 0.1 whenever X is a knownexample of positive sectional curvature, respectively a homogeneous space ofpositive curvature—see Section 1.2 for more details on these classes. Thisis particularly interesting for different reasons: First of all these spacesconstitute a nice class of highly important geometric examples. Second,maybe more strikingly, let us recall the Petersen–Wilhelm conjecture whichstates that whenever X → B is a Riemannian submersion with X (andconsequently B ) positively curved Riemannian manifolds, then 2 dim B > dim X ; respectively, in the case of compact spaces, when this submersion isa fibration F (cid:44) → X → B , dim B > dim F . We recall the general property(for example see [1], [7, Proposition 1, p. 5]) that with X being elliptic (andnot necessarily a manifold with curvature bound) the fibration features inthe category of elliptic spaces already. In particular, this would provide an apriori weaker formulation of Conjecture 0.1.In [1] we proved the Petersen–Wilhelm conjecture in a much more generalcontext for the known examples of positive curvature of even dimensionsonly using their rational structure. (Since several odd-dimensional examplesrationally split as a product, rational tools are not enough in odd dimensions;in [7] the odd-dimensional examples are verified using finite coefficients.)Note further that due to the Bott–Grove–Halperin conjecture positivelycurved manifolds should be elliptic. In even dimensions the equally famousHopf conjecture speculates that they have positive Euler characteristic. Thatis, conjecturally the case of even-dimensional positively curved manifoldsshould be completely covered by Theorem C. MANUEL AMANN
In summary, in this context Conjecture 0.1 controls much more complicatedinvariants of fibration decompositions of positively curved manifolds thanmerely dimensions. Given all previous observations and conjectures onemight speculate that
Conjecture.
For a closed simply-connected positively curved manifold M and any fibration F (cid:44) → M → B of simply-connected spaces Estimate (1) iswell-defined and holds true. Viewed from a different angle, we once again observe that positively curvedmanifolds seem to constitute a class of spaces behaving extremely well withrespect to several different topological approaches. We verify this speculationon the known examples.
Corollary F.
Conjecture 0.1 holds true whenever the cohomology algebra H ∗ ( X ) is generated by at most one even-degree and at most one odd-degreeelement. In particular, this is true if X has the rational type of a simply-connected closed homogeneous space of positive sectional curvature respectivelyof any known example of a closed manifold admitting positive sectionalcurvature. The content of this is the observation that any fibration then only involvesformal spaces. We remark that the confirmation of the conjecture for X positively elliptic is yet another corollary of Theorem E as well: If χ ( X ) >
0, by the multiplicativity of the Euler characteristic in fibrations, so are χ ( B ) , χ ( F ) >
0. It is well-known that F -spaces are formal.We leave it to the reader to reformulate our results in larger generality fornilpotent spaces and nilpotent fibrations. Structure of the article.
In Section 1 we discuss some relevant aspectsfrom Rational Homotopy Theory. In Section 2 we note first observationson the conjecture before, in the second part of the section, we elaboratelyconsider and discuss different notions of convergence for families of ellipticspaces. In Section 3 we explain the proof of Theorem A, which is ratherindependent of the following arguments. Section 4 is devoted to the proof ofTheorem C. In particular, there we prove Lemma 4.2, which is central to ourarguments and underlies nearly all further (and partly even previous) work.Finally, in Section 5 we refine and massively extend the previous argumentsin order to provide proofs of Theorems D and E. As an application of theobtained results we use this to show Corollary F in a subsequent step.
Acknowledgements.
The author was supported both by a Heisenberg grantand his research grant AM 342/4-1 of the German Research Foundation; heis moreover associated to the DFG Priority Programme 2026.1.
Some tools from Rational Homotopy Theory
Excerpts from Rational Homotopy Theory.
This section cannotprovide and is not intended to give an introduction to the theory. We expectthe reader to have gained a certain familiarity with necessary concepts for
OMOLOGY VS. HOMOTOPY 5 example from [5] or [6]. We merely recall some tools and aspects which playa larger role in the article.Many computations of the article rely on the theory of (minimal) Sullivanmodels of simply-connected spaces X . Just to recall these are certain com-mutative differential graded algebras (Λ V, d) encoding the rational homotopytype of X with V a positively graded rational vector space and Λ V the tensorproduct on the symmetric algebra of the evenly-graded part V even and theexterior algebra on the oddly-graded part V odd . Moreover, we use relativemodels and models of fibrations as constructed in [5, Proposition 15.5]. Thatis, for a fibration of simply-connected spaces F (cid:44) → E → B and for Sullivanmodels (Λ V, ¯d) of F and (Λ W, d) of B , a model for E is given by a tensorproduct (Λ V ⊗ Λ W, d) where (Λ W, d) is a differential subalgebra, and theprojection induced by W → V, ¯d) of F .We investigate fibrations and their Sullivan models from a cohomologicaland a homotopical point of view. For the associated Serre spectral sequencesee [5, Chapter 18]. For the associated long exact sequence of homotopygroups in terms of models see [5, Section 15(e), p. 214]. In particular, recallthat with the terminology from the last paragraph, the long exact homotopysequence dualises to the exact sequence . . . → W k → H k ( W ⊕ V, d ) → V k d −→ W k +1 → . . . with transgression d with respect to which also cohomology H k ( W ⊕ V, d )is taken. This d denotes the linear part of the differential d on (Λ( V ⊕ W )defined by im (d − d ) ⊆ Λ ≥ ( V ⊕ W ). (Clearly, V k ∼ = π k ( F ) ⊗ Q , W k ∼ = π k ( B ) ⊗ Q —see [5, Theorem 15.11, p. 208]—and H k ( W ⊕ V, d ) ∼ = π k ( E ) ⊗ Q taking into account that the model of the fibration is not necessarily minimal.)We shall speak of rational homotopy groups of F and B being contracted when passing to X which is supposed to indicate that such a homotopygroup lies in the kernel respectively the image of d and hence exists in F respectively B , but no longer contributes non-trivial homotopy to X .We shall moreover draw on Euler and homotopy Euler characteristics. Weuse the convention to define the latter for an elliptic minimal Sullivan algebra(Λ V, d) as χ π (Λ V, d) = dim V odd − dim V even The Euler characteristic is multiplicative (which can be proved using theSerre spectral sequence), the homotopy Euler characteristic is additive infibrations (as follows from the depicted long exact homotopy sequence).The formal dimension n of an elliptic space, i.e. the largest degree withnon-trivial cohomology can be computed via the following dimension formula using the degrees and dimensions of its homotopy groups—see [5, Theorems32.2 (iii), p. 436 and 32.6 (i), p. 441]. For this we recall the even and oddexponents a i and b i of a minimal Sullivan algebra (Λ V, d) defined by theproperty that the 2 a i are the degrees of a basis of V even and the 2 b i − V odd . It then holds that (cid:88) (2 b i − − (cid:88) (2 a i −
1) = n Compare Remark 4.1.
MANUEL AMANN
Elliptic spaces X of positive Euler characteristic, so-called F -spaces or pos-itively elliptic spaces possess a very rigid structure: Their rational cohomologyis concentrated in even degrees; actually it is given by a polynomial algebramodulo a regular sequence whence these spaces are (hyper-/intrinsically)formal. Moreover, from [5, Formula (32.14), p. 446] we recall that the totaldimension of their cohomology, i.e. their Euler characteristic, which equalsthe sum of all Betti numbers in this case, is given bydim H ∗ ( X ) = q (cid:89) i =1 b i a i (2)where the b i and a i range over the odd respectively the even exponents of aminimal model of X .As a consequence, positively elliptic spaces admit pure models —see [5,Chapter 32, p. 434]. This contributes to the importance of pure spaces inrational homotopy theory. There are many prominent classes of pure spacesfeaturing homogeneous spaces and biquotients as well as cohomogeneity onemanifolds. Recall our definition of two-stage spaces in the introduction whichclearly constitutes a slight generalisation of pureness. Two-stage spaces gainspecial importance due to the following Proposition 1.1 (see Proposition 5.10, p. 32, in [3], cf. [10]) . Let X bea formal elliptic space. Then rationally it is the total space of a totallynon-homologous to zero fibration with model (Λ B, → (Λ B ⊗ Λ V, d) → (Λ V, ¯d) where B = B odd , and (Λ V, ¯d) is positively elliptic. That is, in particular, formal elliptic spaces are two-stage.Recall that a fibration
F (cid:44) → X → B is called totally non-homologous tozero if the induced map H ∗ ( X ) → H ∗ ( F ) is surjective, or, equivalently, ifthe associated Serre spectral sequence degenerates at the E -term. Remark 1.2.
Moreover, we may assume that the model (Λ B ⊗ Λ V, d) is minimal , i.e. the fibration to be π -trivial as well, (and then decompose it asdepicted). This follows from the proof of [3, Proposition 5.10] in which wedecomposed a two-stage model of X with stage one mapping to a regularsequence in the algebra generated by stage 0 in this form. Without restriction,we may choose the model we start with to be minimal. Indeed, the modelcomes from [4, Theorem II, p. 577], and can be chosen as a minimal modelof the hyperformal cohomology H ∗ ( X ). (cid:30) One of the most famous and most influential conjectures in the area is
Conjecture 1.3 (Halperin) . Let
F (cid:44) → X → B be a fibration of simply-connected spaces with F positively elliptic. Then the fibration is totallynon-homologous to zero. In particular, this conjecture was verified • on compact homogeneous spaces of positive Euler characteristic ([19]). • on simply-connected Hard-Lefschetz spaces ([15]). OMOLOGY VS. HOMOTOPY 7 • in the case of at most three generators of the cohomology algebra(see [20] and [11]). • for spaces of formal dimension at most 16 or Euler characteristic atmost 16 (see [2, Theorem 11.6]). • in the “generic case” (cf. [18]). • to be closed under fibrations of simply-connected spaces of finite type(cf. [13]), i.e. if both base and fibre satisfy the Halperin conjecture,so does the total space.These classes of examples enrich Theorem D.Note further that it is known that the Halperin conjecture for an ellipticspace F holds if and only if it holds in the category of elliptic spaces, moreprecisely, already for base spaces being odd dimensional spheres (see [12,Theorem 1.5, p. 6], [14]).1.2. Positively curved spaces and their rational structure.
By “pos-itive curvature” we shall always denote positive sectional curvature.The known examples of simply-connected positively curved closed mani-folds are the following (cf. [7]): • the subsequent homogeneous spaces, namely compact rank one sym-metric spaces S n , C P n , H P n , CaP , the Wallach flag manifolds W , W , W , the Aloff–Wallach spaces W p,q , and the Berger spaces B , B . • the biquotients E due to Eschenburg, the family SU (3) (cid:12) S (parametrised by different inclusions) generalising and comprising theAloff–Wallach spaces, the family of Bazaikin spaces SU (5) (cid:12) Sp (2) S in dimension 13 containing B , and • a cohomogeneity one example P of dimension 7 due to Dearricotand Grove–Verdiani–Ziller.Without going into details, collecting the information for example from [23],[1], [7] we derive that all these spaces are formal and elliptic, and, in anycase, the following holds: • If M is even-dimensional, it is positively elliptic. • If M is odd-dimensional, it satisfies χ π ( M ) = 1. It is either rationallya sphere, or its rational cohomology algebra has exactly one generatorin positive even-degree and exactly one in odd-degree.This is the necessary information underlying the proof of Corollary F forpositively curved manifolds (see Property ( ∗ ) and Remark 5.6).We remark further that there is a classification of simply-connected pos-itively curved homogeneous spaces by Wallach and B´erard–Bergery (forexample see [21]) which states that the cited homogeneous examples areactually the only ones. 2. First observations
Fibrations.
Let
F (cid:44) → X → B a fibration of elliptic spaces. We callit π -trivial , if π ∗ ( X ) ⊗ Q = π ∗ ( F ) ⊗ Q ⊕ π ∗ ( B ) ⊗ Q , or, equivalently, if therelative minimal model of the fibration is actually a minimal model of X . MANUEL AMANN
It is interesting to observe that π -trivial fibrations play a role converse tothe one of totally non-homologous to zero ones with respect to Conjecture0.1; more precisely, • if the fibration is totally non-homologous to zero, then h ( X ) ≤ h ( F ) + h ( B ) (note that the computations of [22, Page 3] for theproduct fibration apply similarly to yield this inequality), and theright hand side of (1) is satisfied, in particular. • if the fibration is π -trivial, then h ( F × B ) ≤ h ( X ), and the left handside of (1) is satisfied, in particular.As for the latter, it suffices to recall that the Serre spectral sequence ofthe fibration F (cid:44) → X → B of simply-connected spaces of finite-dimensionalrational cohomology satisfies E p,q = H p ( B ) ⊗ H q ( F ), whencedim H ∗ ( X ) ≤ dim H ∗ ( F ) · dim H ∗ ( B )(3)If the fibration is π -trivial, it follows that π ∗ ( F × E ) ⊗ Q = π ∗ ( X ) ⊗ Q . Wededuce the given estimate for h ( X ). See also [22, Proposition 3.2, p. 3] wherethe same arguments are used to verify the conjecture for fibrations whichare both π -trivial and totally non-homologous to zero.If the fibration is not π -trivial, which usually is the case, it is in particularnecessary to understand how much rational homotopy is contracted whenpassing to π ∗ ( X ) ⊗ Q . This is dealt with in Theorems C (in the generalsituation) and D (for positively elliptic fibres). So the situation of π -trivialfibrations, or, more generally, both degeneracy properties of “ π -triviality” and“totally non-homologous to zero”, nicely motivate these further generalisations.2.2. Convergence.
There are several notions possible for defining “conver-gence” of a family of elliptic spaces. Let us start discussing them.
Definition 2.1.
Let X be a family of elliptic spaces. We say that X has accumulation point c ∈ R ∪ {∞} if for any ε > X ∈ X with | h ( X ) − c | < ε . We say that X converges to c ∈ R if c is itsonly accumulation point. Example 2.2. • Clearly, by definition, no finite family X of ellipticspaces can have accumulation points nor converge. No family withuniversally bounded cohomology can converge to zero. • Every infinite family X of elliptic spaces has a (possibly infinite)accumulation point. If X converges to c , then so does any infinitesubfamily Y ⊆ X . • The family { C P n } n ≥ is a family of universally bounded homotopydim π odd ( C P n ) ⊗ Q = 2 converging to zero. Compare Proposition2.6. • The family { S n } n ≥ realises infinitely many rational homotopy groups,each element satisfies dim π ∗ ( X ) ≤ X ∈ X . Clearly h ( X ) ∈{ / , } for X in X , and the family has two accumulation points(although the set { h ( X ) | X ∈ X } is finite and hence does not haveany accumulation points). Odd spheres converge to , even ones to1. OMOLOGY VS. HOMOTOPY 9 • There are infinite families X of elliptic spaces realising only finitelymany Betti numbers (for example, see [6, Chapter 6.2, p. 243]). Hencethese families have positive accumulation points. • Such families can already be found to realise the same cohomologyalgebras (see [17]). • All of these example families only realise finitely many rationalhomotopy groups, hence the accumulation points are positive, butnot infinite. Taking product or more elaborate constructions onemay easily adapt limit points. (cid:30)
The next Proposition generalises our observation on the family of spheres.
Proposition 2.3.
Let X be a family of elliptic spaces, let P denote thefamily of pure spaces, Q the one of two-stage spaces. If P ⊆ X respectively
Q ⊆ X , then any number h ( X ) for X ∈ P respectively for X ∈ Q is anaccumulation point of X . Proof . For every pure respectively two-stage space X we construct aninfinite sequence of pure respectively two-stage spaces X i satisfying h ( X ) = h ( X i ) for all i ≥
1. This pureness/two-stage property will be obvious fromthe construction. This can be done as follows.Recall that pure spaces are two-stage in particular. Let (Λ( V ⊕ V ) , d)be the two-stage decomposition of the minimal model of X . We choose theminimal model in its isomorphism class such that we display V with minimalpossible dimension. Hence, the differential is injective on V and differentialshave a well-defined degree. Let v , . . . , v k be a homogeneous basis of V and v (cid:48) , . . . , v (cid:48) k (cid:48) be a homogeneous basis of V . Up to spatial realisation, itsuffices to construct a two-stage minimal model (Λ W, d) = (Λ( W ⊕ W ) , d)of X i —for the sake of simplicity we suppress the index i in the models. Forthis we construct a homogeneous basis w , . . . , w k of W and w (cid:48) , . . . , w (cid:48) k (cid:48) of W . The w j and w (cid:48) j will be degree shifts of the corresponding v j , v (cid:48) j . Weextend degrees multiplicatively. Hence it remains to definedeg w j := 3 i · j deg w (cid:48) j := 3 i · deg(d v (cid:48) j ) − v (cid:48) j = p j ( v i ) asa polynomial p j in the v i , and we denote by p j ( w i ) the correspondingpolynomial replacing the v i by the w i . Hence setd w j := 0d w (cid:48) j := p j ( w i )which is well-defined by construction. Hence all the X i are well-defined purerespectively two-stage spaces. They are all mutually distinct due to degrees.Then all X i have isomorphic minimal models, however, using isomorphisms not respecting the grading. Indeed, by construction, the isomorphism to(Λ V, d) is induced by the correspondence v i ∼ w i , v (cid:48) i ∼ w (cid:48) i . (For this notethat due to multiplication with 3 i the parity of the basis is preserved.) Inparticular, h ( X i ) = h ( X ) for all i . This proves the result. (cid:3) As a consequence the family of pure or two-stage spaces or any familycontaining them like the family of all elliptic spaces does not converge to anylimit point.Note that the elements in the sequences we constructed in the last proofall had the same rational homotopy groups. It seems more interesting tounderstand what happens if rational homotopy tends to infinity.
Definition 2.4.
A family X of elliptic spaces has π -accumulation point c ∈ R ∪ {∞} , if for all ε > n ( ε ) ∈ N and infinitely many X ∈ X with dim π ∗ ( X ) ⊗ Q ≥ n ( ε ) and | h ( X ) − c | < ε .The family π -converges to c ∈ R ∪ {∞} , i.e.lim dim π ∗ ( X ) ⊗ Q →∞ h ( X ) = c if c is the only π -accumulation point.In the folllowing let us discuss zero as an accumulation point. We needsome preparatory results first.We provide an easy and coarse estimate on the dimension of the cohomologyof a pure space. Note that the important aspect for us is that this estimateis given purely in terms of degrees and dimensions of rational homotopygroups, since the formal dimension d of an elliptic space can be computedjust using this degree information. Lemma 2.5.
Let (Λ V, d) be a pure minimal Sullivan algebra of formaldimension d . Denote by a , . . . , a k the degrees of a homogeneous basis of V even . Then dim H (Λ V, d) ≤ dim V odd − dim V even · (cid:89) ≤ i ≤ k (cid:100) d/a i (cid:101) Proof . Let w , . . . , w k be such a homogeneous basis of V even with deg w i = a i . Consider the rational fibration given by the relative model(Λ (cid:104) v , . . . , v k (cid:105) ⊗ Λ V, d)(4)with fibre (Λ (cid:104) v , . . . , v k (cid:105) ,
0) generated by elements of degrees deg v i = a i · ( (cid:100) d/a i (cid:101) + 1) − v i = w (cid:100) d/a i (cid:101) +1 i .By construction—we chose the v i to map to elements of degree larger thanthe formal dimension d of (Λ V, d) under the differential d—the total spaceactually has the following minimal model up to isomorphism.(Λ (cid:104) v , . . . , v k (cid:105) ⊗ Λ V, d) ∼ = (Λ V, d) ⊗ (Λ (cid:104) v , . . . , v k (cid:105) , V even ⊕ (cid:104) v , . . . , v k (cid:105) ) , d) and fibre (Λ V odd , (cid:81) ≤ i ≤ k (cid:100) d/a i (cid:101) . OMOLOGY VS. HOMOTOPY 11
By the E -term of the associated Serre spectral sequence of this newfibration we deduce thatdim H (Λ V, d) · k = dim H ((Λ V, d) ⊗ (Λ (cid:104) v , . . . , v k (cid:105) , ≤ dim H (Λ V odd , · dim H (Λ( V even ⊕ (cid:104) v , . . . , v k (cid:105) ) , d) ≤ dim V odd · (cid:89) ≤ i ≤ k (cid:100) d/a i (cid:101) The assertion follows. (cid:3)
For the next proposition it would have been enough to work with thewell-known estimate dim H ∗ ( X ) ≤ dim X for an elliptic space (and againto use that formal dimension can be expressed via the degrees of rationalhomotopy groups). As a service to the reader we provided the last lemmawith its concise proof instead. Proposition 2.6.
If the family X of elliptic spaces has as an accumulationpoint, then sup X ∈X { dim π ∗ ( X ) ⊗ Q } = ∞ or sup X ∈X { i ∈ N | π i ( X ) ⊗ Q (cid:54) = 0 } = ∞ In any case formal dimensions are unbounded, i.e. sup X ∈X { dim X } = ∞ Proof . It suffices to show that fixing the rational homotopy groups π ∗ ( X ) ⊗ Q of an elliptic space X , there exists α ∈ N such that dim H ∗ ( X (cid:48) ) ≤ α forall elliptic X (cid:48) satisfying π ∗ ( X (cid:48) ) ⊗ Q = π ∗ ( X ) ⊗ Q . Indeed, this implies thatfor dim H ∗ ( X ) to be unbounded within X (which is clearly necessary foraccumulation point zero), it is required to have infinitely many configurations π ∗ ( X ). That is, either infinitely many homotopy Betti numbers or infinitelymany degrees of rational homotopy groups (or both).In any case the dimension formula (see Section 1.1) for elliptic spaces(together with the observation that the existence of an even-degree basis ele-ment of the rational homotopy groups requires the existence of an additionalodd-degree one of at least twice the degree, see [5, Proposition 32.9]) yieldsthe unboundedness of formal dimensions.So let us show the existence of α . By the odd spectral sequence ([5, Chapter32(b), p. 438]) dim H (Λ V, d) ≤ dim H (Λ V, d σ ) for a minimal Sullivan algebra(Λ V, d) with associated pure one (Λ V, d σ ). Hence, without restriction, wemay assume that the X (cid:48) are pure spaces, and we have to show that fixingrational homotopy groups there are only finitely many dim H ∗ ( X (cid:48) ) for pure X (cid:48) realising the homotopy groups. This follows from Lemma 2.5 in whichwe provide an upper bound on cohomology merely in terms of the degreesand dimensions of the rational homotopy groups. (cid:3) Recall the family of complex projective spaces from Example 2.2 withconstant dimension of rational homotopy groups and diverging cohomology.Here the top degrees of rational homotopy diverge. The family of products ofspheres of a fixed dimension clearly has bounded top homotopical degree and diverging homotopical dimension. Both families π -converge to zero. Thisillustrates that both cases in the proposition really can occur.So we already started to answer Question.
Which accumulation points can be realised by a family of ellipticspaces? Or, much more interestingly, which π -accumulation-points can berealised? Remark 2.7.
Instead of merely looking at limits, we also suggest to havea closer look at the rate of convergence. For example, if convergence isgoverned by n (cid:55)→ n n , then the elements of X satisfy the toral rank conjecture.(Of course, this is a rather restrictive condition.)Clearly, it is well-known (see [6, Theorem 7.13, p. 279]) that the toral rankrk ( X ) of an elliptic X satisfies rk ( X ) ≤ χ π ( X ) ≤ dim π odd ( X ). Thenrk ( X )dim H ∗ ( X ) ≤ dim π ∗ ( X ) ⊗ Q dim H ∗ ( X ) ≤ dim π ∗ ( X ) ⊗ Q dim π ∗ ( X ) ⊗ Q and (ignoring trivial cases of contractible X or vanishing rank)dim H ∗ ( X ) ≥ rk ( X ) · rk ( X )dim π ∗ ( X ) ⊗ Q · dim π ∗ ( X ) ⊗ Q rk ( X ) Since dim π ∗ ( X ) ⊗ Q − rk ( X ) + log rk ( X ) − log dim π ∗ ( X ) ⊗ Q is greaterequal 0 for all relevant values, the toral rank conjecture holds in this situation. (cid:30) Next we investigate how convergence to zero behaves under fibrationswhence extending the class to more instances.
Proposition 2.8.
Let
F IB = (
F (cid:44) → X → B ) be a family of totally non-homologous to zero fibrations of elliptic spaces. • The family X of total spaces π -converges to zero if both the family F of fibres and the family B of base spaces do. • The family of total spaces X is π -convergent to zero if and only if sois the family of spaces F × B = ( F × B ) . Proof . Assume first that both F and B π -converge to zero, and we shallshow that X also π -converges to 0. Now with h ( F ) and h ( B ) also h ( X ) = dim π ∗ ( X ) ⊗ Q dim H ∗ ( X ) ≤ dim π ∗ ( F ) ⊗ Q + dim π ∗ ( B ) ⊗ Q dim H ∗ ( X ) ≤ h ( F ) + h ( B )tends to zero (using the assumption that both dim H ∗ ( F ) , dim H ∗ ( B ) ≤ dim H ∗ ( X )).Due to Lemma 4.2 and the formula3 dim π ∗ ( X ) ⊗ Q ≥ dim π ∗ ( F ) ⊗ Q + dim π ∗ ( B ) ⊗ Q which we obtain from there, we derive that with π ∗ ( F ) ⊗ Q and π ∗ ( B ) ⊗ Q also dim π ∗ ( X ) ⊗ Q is unbounded. Hence X π -converges to zero.Now we deal with the second assertion. By the very last argument wederive that X has unbounded rational homotopy if and only if F × B has. It
OMOLOGY VS. HOMOTOPY 13 remains to show that h ( X ) tends to zero if and only if h ( F × B ) does. Dueto the fibrations being totally non-homologous to zero we have · h ( F × B ) = · dim π ∗ ( F × B ) ⊗ Q dim H ∗ ( X ) ≤ h ( X ) = dim π ∗ ( X ) ⊗ Q dim H ∗ ( X ) ≤ dim π ∗ ( F ) ⊗ Q + dim π ∗ ( B ) ⊗ Q dim H ∗ ( F × B ) = h ( F × B )and the assertion follows. (cid:3) To avoid confusion: for the family
F × B the respective spaces F and B belong to the same fibration. As the proof shows, for the first part of thestatement instead of a totally non-homologous to zero fibration it would beenough to have the weaker properties dim H ∗ ( F ) , dim H ∗ ( B ) ≤ dim H ∗ ( X ).3. Proof of Theorem A
We use the two-stage decomposition for the space X described in theintroduction. Proof of Theorem A . We need to show that for arbitrarily large num-bers there exist infinitely many two-stage spaces X with larger homotopy andwith h ( X ) tending to zero. It is clear (for example just by taking products oftwo-stage spaces) that the class of two-stage spaces has unbounded rationalhomotopy. Hence it remains to see that the number h ( X ) tends to zero withdim π ∗ ( X ) ⊗ Q going to infinity.We recall from [9, Theorem 2.3, p. 195] thatdim H ∗ ( X ) ≥ dim W − dim V even Moreover, by word-lengthdim H ∗ ( X ) ≥ ≤ V even + dim Λ ≤ W + dim V even · dim W − dim W =1 + 2 dim V even + (cid:18) dim V even (cid:19) + dim W + (cid:18) dim W (cid:19) + dim V even · dim W − dim W Set n := dim V even , m := dim W , r := dim W − dim V even . (It is clear that r ≥ h ( X ) ≤ n + m + r max( ( n + n + m + m + 2 nm + 2 − r ) , r )and we have to show that as one of n, m, r goes to infinity, this expressionfalls below 1 /k for any k ∈ N .We consider two different cases. Case 1.
Suppose that r ≤ n + m , which implies that h ( X ) ≤ n + 3 mn + m + 2 nm + 2In this case, whenever n → ∞ or m → ∞ the right hand side becomesarbitrarily small. Case 2.
Suppose that r ≥ n + m , i.e. , in particular, 4 r ≥ n + m . Then r tends to infinity if so do n or m . Moreover, h ( X ) ≤ n + m + r r ≤ r r This converges to 0 whenever any of n, m, r tend to infinity. (cid:3)
We leave it to the reader to make use of the fact that the estimates areexplicit, i.e. to provide concrete numbers for n, m, r for which the estimateshold. 4.
Proof of Theorem C
Remark 4.1.
In the following we shall draw on the dimension formula(see Section 1.1). We remark that this formula for general elliptic Sullivanalgebras (Λ V, d) is obtained by a reduction to the pure case, i.e. by passing tothe associated pure model (Λ V, d σ ). For this (see [5, Proposition 32.4, p. 438])it is shown that in the case when (Λ V, d) is minimal its cohomology is finitedimensional if and only if so is H (Λ V, d σ ). In [5, Proposition 32.7, p. 442] itis shown that (Λ V, d σ ) and (Λ V, d) have the same maximal degrees of non-vanishing cohomology. Although, as it seems, despite not being explicitelyrequired in the assertions the proof of this latter result also assumes theminimality of (Λ V, d). Clearly, already the cohomological finiteness resultis wrong without the minimality assumption, as already the example of thecontractible algebra (Λ (cid:104) x, y (cid:105) , x (cid:55)→ y, deg x = 2 , deg y = 3) with associatedpure algebra (Λ (cid:104) x, y (cid:105) ,
0) of infinite-dimensional cohomology shows. Also[5, Theorems 32.6, p. 441, and 32.9, p. 442] draw on minimality although,putatively, not stated.Clearly, the difference between minimal and non-minimal models is eradi-cated when formulating the dimension formula in terms of homotopy groups(see [5, p. 434]). The dimension formula, however, stays correct the wayit is formulated via even and odd exponents of Sullivan algebras (see Sec-tion 1.1) also for non-minimal algebras if either (Λ V, d) is pure or underthe following restriction: Up to isomorphism a Sullivan algebra can bewritten as the product of a minimal and a contractible one (see [5, The-orem 14.9, p. 187]). Hence it remains to verify when the dimension for-mula holds for the contractible factor, i.e. basically for the two situations(Λ (cid:104) x, y (cid:105) , x (cid:55)→ y ) once for deg x odd and deg y = deg x + 1 even, and once fordeg x even and deg y = deg x + 1 odd. In the first case we obtain dimensiondeg x − (deg y −
1) = deg x − deg x = 0, the dimension formula holds; in thesecond one it fails due to deg y − (deg x −
1) = deg y − deg y + 2 = 2. (Notethat a pureness assumption excludes the second case.)However, in the proof of the following lemma we see that the latter algebracannot be decomposed as the total space of a fibration of elliptic spaces byexactly comparing the dimension formula of such total spaces with the ones OMOLOGY VS. HOMOTOPY 15 of the corresponding product spaces of potential fibre and base. Indeed, thisboils down to exactly the same “(+2)-contradiction” we just observed. (cid:30)
We now prove a crucial lemma underlying several results. Note that wealready drew on it in Section 2 (which we do not use at all for the subsequentreasoning).
Lemma 4.2.
Let
F (cid:44) → X → B be a fibration of rationally elliptic spaces.Then it holds that dim π odd ( X ) ⊗ Q ≥ dim π odd ( B ) ⊗ Q ≥ dim π even ( B ) ⊗ Q dim π odd ( X ) ⊗ Q ≥ dim π even ( X ) ⊗ Q ≥ dim π even ( F ) ⊗ Q dim π odd ( X ) ⊗ Q ≥ dim π odd ( F ) and, in total, dim π ∗ ( X ) ⊗ Q + 2 dim π odd ( X ) ⊗ Q ≥ dim π ∗ ( F ) ⊗ Q + dim π ∗ ( B ) ⊗ Q Proof . We recall the dimension formula (see Section 1.1) for the rationallyelliptic space X . dim X = (cid:88) i b i − (cid:88) j ( a j − b i range over the degrees of a homogeneous basis of π odd ( X ) ⊗ Q ,and the analog for the a j and π even ( X ) ⊗ Q , and dim X denotes formaldimension.We now fix models and homogeneous bases of base and fibre, namely(Λ (cid:104) f i (cid:105) i , ¯d) a minimal model of F , and (Λ (cid:104) b j (cid:105) j , d) one of B yielding the modelof the fibration, i.e. a (not necessarily minimal) Sullivan model for X givenby (Λ (cid:104) f i , b j (cid:105) i,j , d)Consider the long exact homotopy sequence π i ( F ) ⊗ Q → π i ( X ) ⊗ Q → π i ( B ) ⊗ Q ∂ −→ π i − ( F ) ⊗ Q → π i − ( X ) ⊗ Q Up to a change of basis we may assume that (passing to the dual sequence)ker ∂ ∗ = (cid:104) f i (cid:105) ≤ i ≤ m . Consequently (see Section 1.1 and the description of thedifferential there),d | (cid:104) f i (cid:105) ≤ i ≤ m : (cid:104) f i (cid:105) ≤ i ≤ m → Λ (cid:104) b i , f j (cid:105) i,j / Λ ≥ (cid:104) b i , f j (cid:105) i,j ∼ = (cid:104) b i , f j (cid:105) i,j is injective with image in (cid:104) b i (cid:105) i . Again, up to change of basis, we may assumethat d ( f i ) = b i , and deg f i + 1 = deg b i for 1 ≤ i ≤ m . Hence a minimalmodel of X is given by (Λ (cid:104) f i , b j (cid:105) i,j>m , ˜d) with a suitably adapted differential˜d. Next, we use the equality of formal dimensions dim X = dim F + dim B (which can easily be deduced from the Serre spectral sequence and thefact that E dim B, dim F = Q , which is left invariant by the differentials), andcompute both sides separately. By applying the dimension formula to the two respective minimal models of X and of F × B it follows that (cid:88) i>m deg b odd i + deg f odd i − (cid:88) j>m (deg b even j + deg f even j − (cid:88) i deg b odd i + deg f odd i − (cid:88) j (deg b even j + deg f even j − (cid:88) i ≤ m deg b odd i + deg f odd i − (cid:88) j ≤ m (deg b even j + deg f even j − (cid:88) i ≤ m (deg f even i + 1) − (deg f even i −
1) + (cid:88) i ≤ m deg f odd i − (cid:0) deg f odd i + 1 − (cid:1) = 2 · ≤ i ≤ m f even i It follows that there is no even-degree element in the kernel of d = ∂ ∗ ,i.e. any even-degree rational homotopy group of F passes non-trivially to X .Respectively, the equation writes as0 = (cid:88) i ≤ m deg b odd i + deg f odd i − (cid:88) j ≤ m (deg b even j + deg f even j − (cid:88) i ≤ m deg b odd i − (deg b odd i − −
1) + (cid:88) i ≤ m (deg b even i − − (cid:0) deg b even i − (cid:1) = 2 · ≤ i ≤ m b odd i and odd-degree rational homotopy groups of the base space B pass injectivelyto X . Both observations taken together prove thatdim π even ( X ) ⊗ Q ≥ dim π even ( F ) ⊗ Q anddim π odd ( X ) ⊗ Q ≥ dim π odd ( B ) ⊗ Q It is well known (see [5, Proposition 32.10, p. 444]) that a rationally elliptic Y satisfies dim π odd ( Y ) ⊗ Q ≥ dim π even ( Y ) ⊗ Q . Hence it remains to provethat dim π odd ( X ) ⊗ Q ≥ dim π odd ( F ) ⊗ Q whence the formuladim π ∗ ( X ) ⊗ Q + 2 dim π odd ( X ) ⊗ Q ≥ dim π ∗ ( F ) ⊗ Q + dim π ∗ ( B ) ⊗ Q follows by summation.The linear part of the differential d, namelyd : (cid:104) f odd i (cid:105) i → (cid:0) Λ (cid:104) f i , b j (cid:105) i,j / Λ ≥ (cid:104) f i , b j (cid:105) i,j (cid:1) even ∼ = (cid:104) f even i , b even j (cid:105) i,j maps into (cid:104) b even i (cid:105) i . From the proof on [5, p. 443] we cite that for each b even i there exists a basis element b odd i (of degree at least 2 deg b even i − | (cid:104) f odd i (cid:105) i passes directly to π odd ( X ) ⊗ Q , and that also itsimage, im d , is injectively represented in the odd-degree rational homotopyof X . The intersection of those odd degree elements contributed by the fibreand those by the base is clearly trivial. It follows thatdim π odd ( F ) ⊗ Q = dim (cid:104) f odd i (cid:105) i = dim ker d | (cid:104) f odd i (cid:105) i + dim im d | (cid:104) f odd i (cid:105) i ≤ dim π odd ( X ) ⊗ Q (cid:3) OMOLOGY VS. HOMOTOPY 17
Remark 4.3.
We remark that the estimatedim π ∗ ( X ) ⊗ Q + 2 dim π odd ( X ) ⊗ Q ≥ dim π ∗ ( F ) ⊗ Q + dim π ∗ ( B ) ⊗ Q is sharp as is shown by the example of the Hopf fibration S (cid:44) → S → S . (cid:30) We are finally in the position to provide the
Proof of Theorem C . From (3) we recall that dim H ∗ ( X ) ≤ dim H ∗ ( F ) · dim H ∗ ( B ). It follows from Lemma 4.2 that for elliptic spaces F (cid:44) → X → B the following estimate holds: h ( F × B ) = dim π ∗ ( F ) ⊗ Q + dim π ∗ ( B ) ⊗ Q dim H ∗ ( F ) · dim H ∗ ( B ) ≤ π ∗ ( X ) ⊗ Q dim H ∗ ( X ) = 3 h ( X )(5) (cid:3) Proofs of Theorems D and E
We shall now refine previous arguments to the case of positively elliptic F or X . Proof of Theorem D . Let us first prove the right Inequality in (1) inthe depicted cases. For this we observe the following: Due to the multiplica-tivity of the Euler characteristic in fibrations (see Section 1.1), the space X is F if and only if so are both F and B . Hence if X is F so are allspaces involved. Moreover, a positively elliptic space has rational cohomologyconcentrated in even degrees (see [5, Proposition 32.10]). Hence the Serrespectral sequence degenerates for lacunary reasons, and the fibration is totallynon-homologous to zero whence the right inequality in (1) holds (see Section2.1).The degeneration at the E -term is enforced by the assumption that F satisfies the Halperin conjecture.Let us now deal with the left inequality in (1). We observed that in bothsettings from the assertion F is positively elliptic. As in the proof of TheoremC, we recall that dim H ∗ ( X ) ≤ dim H ∗ ( F ) · dim H ∗ ( B ). From Inequality (5)we recall that h ( F × B ) ≤ h ( X ). In the case when F is positively elliptic weimprove this to h ( F × B ) ≤ h ( X ) by refining the respective proof. Indeed,it now suffices to show that2 dim π ∗ ( X ) ⊗ Q ≥ dim π ∗ ( F ) ⊗ Q + dim π ∗ ( B ) ⊗ Q (6)since then h ( F × B ) = dim π ∗ ( F ) ⊗ Q + dim π ∗ ( B ) ⊗ Q dim H ∗ ( F × B ) ≤ · · (dim π ∗ ( F ) ⊗ Q + dim π ∗ ( B ) ⊗ Q )dim H ∗ ( X ) ≤ · dim π ∗ ( X ) ⊗ Q dim H ∗ ( X )= 2 · h ( X ) (Note that the first inequality is actually an equality using that our fibrationis totally non-homologous to zero; yet, this is irrelevant for the argument atthis stage of the proof.)As we observed in the proof of Lemma 4.2, (im d ) odd = 0, i.e. only odddegree homotopy groups from F contract even degree ones from B . Thisimplies thatdim π ∗ ( X ) ⊗ Q = dim π ∗ ( F ) ⊗ Q + dim π ∗ ( B ) ⊗ Q − c where c ≤ min { dim π odd ( F ) ⊗ Q , dim π even ( B ) ⊗ Q } Since both F is an F -space, we derive thatdim π odd ( F ) ⊗ Q = dim π even ( F ) ⊗ Q dim π ∗ ( F ) ⊗ Q = 2 dim π odd ( F )Since χ π ( B ) ≥
0, we always have for elliptic B thatdim π odd ( B ) ⊗ Q ≥ dim π even ( B ) ⊗ Q dim π ∗ ( B ) ⊗ Q ≥ π even ( B )It follows thatdim π ∗ ( X ) ⊗ Q ≥ dim π ∗ ( F ) ⊗ Q + dim π ∗ ( B ) ⊗ Q − min { dim π ∗ ( B ) ⊗ Q , dim π ∗ ( F ) ⊗ Q }≥ max { dim π ∗ ( B ) ⊗ Q , dim π ∗ ( F ) ⊗ Q } whence Inequality (6). (cid:3) Remark 5.1.
The inequality (6) certainly does not hold when B is positivelyelliptic (instead of F ). For this just consider the Hopf fibration S (cid:44) → S → S (with corresponding inequality 2 <
3) again. (cid:30)
In the proof of Theorem D we came to a point where we had to discussfibrations of F -spaces. Those are necessarily totally non-homologous to zero.In the proof of Theorem E we have to deal with fibrations of formal ellipticspaces. As F -spaces are formal, this generalises the previous discussion.However, such a fibration is no longer necessarily totally non-homologous tozero, as again the example of the Hopf fibration S (cid:44) → S → S shows already.Hence we shall have to discuss the trade-off of homotopy and cohomologydegeneration.Let F (cid:44) → X → B be a fibration of formal elliptic spaces. Due to Propo-sition 1.1 we know that such a formal elliptic space has the structure ofthe total space of a totally non-homologous to zero fibration of an F -spaceover a product of odd-dimensional spheres. Hence X admits the followingSullivan model (Λ V F ⊗ Λ T F ⊗ Λ V B ⊗ Λ T B , d)with T F , T B concentrated in odd degrees, V even B = V odd B , V even F = V odd F ,(Λ V F ⊗ Λ T F , ¯d) a model of F , (Λ V B ⊗ Λ T B , d) a model of B . Next we provethat whenever we contract an element of T F cohomology halves at least. OMOLOGY VS. HOMOTOPY 19
Lemma 5.2. dim H ∗ ( X ) ≤ dim H ∗ ( F × B )2 dim im (d | TF ) Proof . We denote by2 a , . . . , a dim V even F , a dim V even F +1 , . . . , a dim V even F + V even B the degrees of a homogeneous basis of V even F ⊕ V even B , and by2 b − , . . . , b dim V odd F − , b dim V odd F +1 − , . . . , b dim V odd F +dim V odd B − , b dim V odd F +dim V odd B +1 − , . . . , b dim V odd F +dim V odd B +dim T F − , b dim V odd F +dim V odd B +dim T F +1 − , . . . , b dim V odd F +dim V odd B +dim T F +dim T B − V odd F ⊕ V odd B ⊕ T F ⊕ T B .Since X is formal as well, we can compute its cohomology using the degreesof the rational homotopy groups of F and B . Thus it holds thatdim H ∗ ( X ) = 2 dim T F +dim T B · (cid:89) ≤ i ≤ dim V even B +dim V even F b π ( i ) /a i (7)for some permutation π of { , . . . , dim V odd F + V odd B + dim T F } in particularsatisfying b π ( i ) ≤ b i . The first factor comes from the product of odd spheresover which X (being formal elliptic) fibres rationally and in a totally non-homologous to zero manner; actually dim T F + dim T B = χ π ( X ). The righthand side computes the cohomological dimensions of possible positively ellip-tic fibre parts of this totally non-homologous to zero fibration decompositionof X . For this we observe that odd-degree homotopy of this part a priorimay come from all of V odd F ⊕ V odd B ⊕ T F . The degree restrictions for the b i essentially draw on this factor being positively elliptic, i.e. a non-trivialrelation of lower degree cannot be replaced by one of higher degree whencethe one of higher degree must be trivial and yields a free factor of odddegree—indeed, the number of relations in the positively elliptic part equalsthe number of cohomology generators.As we need to take into account that some homotopy groups might becontracted, we may even have that b π ( i ) = a i . We shall make this moreprecise. Let c := dim im (d | T F ) denote the dimension of the subgroup of π even ( B ) ⊗ Q which is contracted by the rational homotopy groups of F dualto T F and hence does not contribute to π even ( X ) ⊗ Q . (We focus on thishomotopy solely, although, clearly, more homotopy groups may be contractedby means of d ( V odd F ).) We may express the F -part in the previous estimateas (cid:89) ≤ i ≤ dim V even B +dim V even F b π ( i ) /a i = (cid:89) ≤ i ≤ dim V even B +dim V even F − c b π ( i ) /a i (cid:124) (cid:123)(cid:122) (cid:125) ≥ · (cid:89) dim V even B +dim V even F − c +1 ≤ i ≤ dim V even B +dim V even F b π ( i ) /a i (cid:124) (cid:123)(cid:122) (cid:125) =10 MANUEL AMANN where we reordered such that the last c factors are those contracted by T F as depicted. For this, recall again from [5, p. 443] that, after decomposingthe algebra into a minimal one times a contractible one, up to reordering thequotients b π ( i ) /a i are at least 2 on the minimal factor; they equal 1 on thecontractible factor, since, from the proof of Lemma 4.2 we recall that d istrivial on V even F , and only an odd-degree element of V F can map non-triviallyto V B . Hence the dimension formula yields b π ( i ) /a i = 1 in this situation.Next we draw some consequences from this description: The minimalmodel of F × B is just the product of the minimal models of F and B .Hence, in order to compute its cohomology, every factor b i /a i ≥ ≤ i ≤ dim V even B + dim V even F yields a factor of at least 2. In other words,every basis element of T F contracted via d hence reduces the dimension ofthe cohomology of F × B by a factor of 2 at least. (Note that this is not truefor elements contracted by d ( V odd F ).) This together with b π ( i ) ≤ b i fromabove implies thatdim H ∗ ( X ) ≤ dim T F +dim T B · (cid:89) ≤ i ≤ dim V even F +dim V even B − c b π ( i ) /a i ≤ dim T F +dim T B − c · (cid:89) ≤ i ≤ dim V even F +dim V even B b i /a i =2 − c · dim H ∗ ( F × B )which proves the asserted estimate. (cid:3) This now enables us to prove Theorem E in the form of the next twopropositions, one for each estimate in (1).
Proposition 5.3.
Let
F (cid:44) → X → B be a fibration of formal elliptic spaces.Then h ( F × B ) ≤ · h ( X ) . Proof . We recall from Lemma 4.2 thatdim π ∗ ( F × B ) ⊗ Q ≤ π ∗ ( X ) ⊗ Q We combine this with Lemma 5.2 and the notation from its proof (inparticular, c = dim im (d | T F )) leading to h ( X ) = dim π ∗ ( X ) ⊗ Q dim H ∗ ( X ) ≥ c · dim π ∗ ( X ) ⊗ Q dim H ∗ ( F × B ) ≥ c · (cid:0) · (dim π ∗ ( F × B ) ⊗ Q ) (cid:1) dim H ∗ ( F × B )= 2 c · · h ( F × B )Hence, in order to establish h ( F × B ) ≤ h ( X ), it remains to observe that2 c +1 · ≥ c ≥ c = 0.If c = 0, we argue as follows. We decompose the minimal model of F as(Λ V F ⊗ Λ T F , ¯d) as in the proof of Lemma 5.2. Hence by the arguments fromthe proof of Lemma 4.2, d can only be non-trivial on a space of dimensiondim V odd F . Clearly, dim V odd F ≤ dim( V F ⊕ T F ). Analogously, im d ⊆ V even B , OMOLOGY VS. HOMOTOPY 21 and dim im d ≤ dim( V B ⊕ T B ). That is, both from fibre and from basespace at most half-dimensional rational homotopy is contracted. Hencedim π ∗ ( X ) ⊗ Q ≥
12 dim π ∗ ( F × B )Adapting the inequalities above and using (3), we then have h ( X ) = dim π ∗ ( X ) ⊗ Q dim H ∗ ( X ) ≥ · dim π ∗ ( F × B ) ⊗ Q dim H ∗ ( F × B )= · h ( F × B )and the result follows also in this case. (cid:3) Proposition 5.4.
Let
F (cid:44) → X → B be a fibration of formal elliptic spaces.Then h ( X ) < h ( F ) + h ( B ) + 14 Proof . The proof basically consists of refining Equation (7)—in particular,drawing on the terminology and results established there. Hence we recallthat dim H ∗ ( X ) = 2 dim T F +dim T B · (cid:89) ≤ i ≤ dim V even B +dim V even F b π ( i ) /a i for some permutation π of { , . . . , dim V odd F + V odd B + dim T F } satisfying b π ( i ) ≤ b i .We now claim and prove that up to renumbering ( π (1) , . . . , π (dim V even F )) =(1 , . . . , dim V even F ), i.e. the first dim V even F many b i come from the F -part ofthe fibre F , i.e. they are given by the fact that the 2 b i − V odd F .In order to prove this, we again draw on the observations from [5, p. 443]respectively on [5, Proposition 32.9, p. 442] and its proof. That is, we haveseen that V even F corresponds injectively (respecting degrees) to a subspace of π even ( X ) ⊗ Q . Hence in the (not necessarily minimal) model of the fibrationthere must exist a homogeneous subspace S of V odd F ⊕ V odd B ⊕ T F ⊕ T B ofdimension at least dim V even F with the property that ¯d S ⊆ Λ V even F and that(Λ V even F ⊗ Λ S, d) is elliptic. Here, as usual, ¯d denotes the projection of d to thefibre Λ( V F ⊕ T F ). Since ¯d | T F ⊕ V B ⊕ T B = 0, the only such subspace is gradedlyisomorphic to V odd F itself. In other words, since dim V odd F = dim V even F ,these first degrees b i for 1 ≤ i ≤ dim V even F are uniquely determined by ahomogeneous basis of V odd F .Hence using (2) we can refine Formula (7) bydim H ∗ ( X ) = 2 dim T B · dim H ∗ ( F ) · (cid:89) ≤ i ≤ dim V even B b π ( i ) /a i (8)where the b i are the odd exponents of V B ⊕ T F , and π now is a permutationof { , . . . , V odd B + dim T F } satisfying b π ( i ) ≤ b i . Indeed, we have seen thatboth even and also odd exponents of the F -part of F appear in the product.That is, the product of their quotients computes the dimension of the cohomology of the F -part of F , i.e. dim H (Λ V F , ¯d). With the decompositionof formal elliptic spaces (see Proposition 1.1 and Remark 1.2) yieldingdim H ∗ ( F ) = dim H ( T F , · dim H (Λ V F , ¯d), and with 2 dim T F = dim H ( T F )the refined formula follows.We derive the estimatedim H ∗ ( X ) ≥ dim T B +dim V even B · dim H ∗ ( F )or, equivalently,dim H ∗ ( X ) ≥ χ π ( B )+dim π even ( B ) ⊗ Q · dim H ∗ ( F )(9)Clearly, we have that χ π ( B ) + dim π even ( B ) ⊗ Q ≥ · dim π ( B ) ⊗ Q , anddim π ∗ ( X ) ⊗ Q ≤ dim π ∗ ( F ) ⊗ Q + dim π ∗ ( B ) ⊗ Q .Hence we can estimate h ( X ) = dim π ∗ ( X ) ⊗ Q dim H ∗ ( X ) ≤ dim π ∗ ( X ) ⊗ Q χ π ( B )+dim π even ( B ) ⊗ Q · dim H ∗ ( F ) ≤ dim π ∗ ( F ) ⊗ Q + dim π ∗ ( B ) ⊗ Q (dim π ∗ ( B ) ⊗ Q ) / · dim H ∗ ( F )= dim π ∗ ( F ) ⊗ Q (dim π ∗ ( B ) ⊗ Q ) / · dim H ∗ ( F ) + dim π ∗ ( B ) ⊗ Q (dim π ∗ ( B ) ⊗ Q ) / · dim H ∗ ( F )(10)The formula h ( X ) < h ( F ) + h ( B ) + trivially holds true whenever oneof F and B are contractible. Hence we may assume this not to be thecase. As an elliptic space satisfies Poincar´e duality, it follows that bothdim H ∗ ( F ) , dim H ∗ ( B ) ≥
2. We derive that h ( X ) < h ( F ) + dim π ∗ ( B ) ⊗ Q (dim π ∗ ( B ) ⊗ Q ) / (11)and we need to discuss the inequality n n/ ≤ for n ∈ N (playing the roleof dim π ∗ ( B ) ⊗ Q ). This holds true unless n ∈ [1 , H ∗ ( F ) = 2, implying that either(1) F (cid:39) S k +1 , k ≥
0, or(2) F (cid:39) S k , k ≥ H ∗ ( F ) = 3 equivalent to F (cid:39) Q Q [ x ] /x .(iii) dim H ∗ ( F ) ≥ Case (i.1).
Let us first deal with Case (i.1). That is, we consider afibration S (cid:44) → X → B with dim π ∗ ( B ) ⊗ Q ≤
7. It follows that dim π ∗ ( X ) ⊗ Q ≤
8. As X is formal,from Proposition 1.1 we derive that, depending on the dimension of itsrational homotopy, the cohomology of X satisfies the following: (dim π ∗ ( X ) ⊗ Q , H ∗ ( X )) can be estimated from below by ( n, ≥ (cid:100) n/ (cid:101) ). Correspondingly, OMOLOGY VS. HOMOTOPY 23 in the respective cases, (dim π ∗ ( X ) ⊗ Q , h ( X )) can be estimated by (1 , ≤ ),(2 , ≤ , ≤ ), (4 , ≤ , ≤ ), (6 , ≤ ), (7 , ≤ ), (8 , ≤ ).Since in our case h ( F ) = h ( S k +1 ) = 1 /
2, we derive that the inequality h ( X ) ≤ / / / h ( X ) 16) for (dim π ∗ ( X ) ⊗ Q , H ∗ ( X )) and by (2 , ≤ ),(4 , ≤ ) for (dim π ∗ ( X ) ⊗ Q , h ( X )). Hence we are also done in these cases.We remark that the arguments underlying this are our usual estimatesof the cohomology of the F -factor: given the decomposition (Λ B ⊗ Λ V, d)from Proposition 1.1 and Remark 1.2 we estimate dim H (Λ B, ≥ dim B and dim H (Λ V, d) ≥ dim V/ . That is, once we have fixed the dimensiondim π ∗ ( X ) = dim V + dim B (see Remark 1.2) of the total rational homotopy,the smaller the dimension of the rational homotopy of the F -part, dim V ,the larger the overall cohomology dim H ∗ ( X ) predicted by this estimate. Case (i.2). Case (ii). Since the Halperin conjecture is confirmed forspaces with cohomology algebra generated by one element (see Section 1.1;for monicly generated cohomology this is just a trivial computation), thefibration is totally non-homologous to zero in Cases (i.2) and (ii). As werecalled in Section 2.1, the formula h ( X ) ≤ h ( F ) + h ( B ) holds wheneverthe fibration is totally non-homologous to zero. Hence we are done in thesecases. Case (iii). In Case (iii) we refine Inequality (11) to h ( X ) < h ( F ) + dim π ∗ ( B ) ⊗ Q dim π ∗ ( B ) ⊗ Q / (12)and solve that n n/ ≤ holds for n ∈ N \ { } . Hence, eventually, assumedim π ∗ ( B ) ⊗ Q = n = 3. Let us see that this case is merely an artefact of theproof. Indeed, we provide a refined version of Estimate (10) directly derivedusing Inequality (9) and not the simplification χ π ( B ) + dim π even ( B ) ⊗ Q ≥ · dim π ( B ) ⊗ Q . That is, we obtain h ( X ) < h ( F ) + dim π ∗ ( B ) ⊗ Q χ π ( B )+dim π even ( B ) ⊗ Q · dim H ∗ ( F )The cases when dim π ∗ ( B ) ⊗ Q = n = 3 now are the following: • Either χ π ( B ) = 3 and, due to formality, B rationally is a product ofthree odd-dimensional spheres, or • χ π ( B ) = 1 and B has an F -component with cohomology algebragenerated by one element.In the first case respectively the second case we derive thatdim π ∗ ( B ) ⊗ Q χ π ( B )+dim π even ( B ) ⊗ Q · dim H ∗ ( F ) ≤ · < π ∗ ( B ) ⊗ Q χ π ( B )+dim π even ( B ) ⊗ Q · dim H ∗ ( F ) ≤ · < and we are done. (cid:3) As a corollary of the proof let us fix Observation (9) again, as it may beof independent interest. Corollary 5.5. For a fibration of formal elliptic spaces F (cid:44) → X → B wehave the estimate dim H ∗ ( X ) ≥ χ π ( B )+dim π even ( B ) ⊗ Q · dim H ∗ ( F ) (cid:3) We sum up the results of these propositions. Proof of Theorem E . Clearly, Theorem E is a combination of Proposi-tions 5.3, and 5.4. (cid:3) We finally prove the Conjecture for an interesting class of manifolds, namelyfor any fibration with X rationally one of the known simply-connectedmanifolds of positive sectional curvature. In particular, this includes allsimply-connected homogeneous spaces admitting homogeneous metrics ofpositive curvature—see Section 1.2 for details. The key observation for thisis that any such manifold M is formal and has the rational structure of an F -space, if dim M is even. If dim M is odd, they satisfydim π odd ( M ) ⊗ Q = 2 and dim π even ( M ) ⊗ Q = 1( ∗ )unless M rationally is an odd-dimensional sphere. Remark 5.6. Indeed, clearly, the class of spaces with finite dimensionalrational cohomology and satisfying ( ∗ ) is exactly the class of spaces withrational cohomology algebra generated by one even-degree and one odd-degreeelement. (cid:30) Lemma 5.7. An elliptic Sullivan algebra (Λ V, d) satisfying ( ∗ ) is formal. Proof . A minimal model of this algebra is of the form(Λ V, d) = (Λ (cid:104) x, y, z (cid:105) , d) with deg x even, deg y , deg z odd. Since the as-sociated pure algebra has finite-dimensional cohomology if and only if theoriginal one has, we derive that, without restriction, deg z ≥ deg y anddeg z > deg x . We derive that (Λ V, d) decomposes as the total space ofa fibration with fibre (Λ (cid:104) x, z (cid:105) , ¯d) over (Λ (cid:104) y (cid:105) , V, d) is simply-connected whence deg y > 1, we obtain that d x = 0. Consequently,(Λ V, d) ∼ = (Λ (cid:104) x, z (cid:105) , d) ⊗ (Λ (cid:104) y (cid:105) , 0) with d x = 0 and d z = x k for some k > (cid:3) Proof of Corollary F . We need to discuss the potential fibrations forall total spaces X which we depicted around ( ∗ ). Case 1. If X is even-dimensional, then X is an F -space. Hence theresult follows from Theorem D. Case 2. Let us assume that X rationally is an odd-dimensional sphere.By Lemma 4.2 we obtain that dim π even ( F ) ⊗ Q = 0, dim π odd ( B ) ⊗ Q ≤ π even ( B ) ⊗ Q ≤ π odd ( F ) ⊗ Q ≤ 1. In total, it followsthat both F and B have one of the models (Λ (cid:104) x (cid:105) , x odd, or (Λ (cid:104) x, y (cid:105) , d) OMOLOGY VS. HOMOTOPY 25 with deg x even, deg y odd, d x = 0, d y = x k for some k > 1. Both are formal,and the result follows from Propositions 5.3 and 5.4. Case 3. Now suppose X is odd-dimensional and not a sphere. Then ( ∗ )applies. By the additivity of the homotopy Euler characteristic we know that1 = χ π ( M ) = χ π ( F ) + χ π ( B ). Hence either χ π ( F ) = 0 and χ π ( B ) = 1 or χ π ( F ) = 1 and χ π ( B ) = 0. In the first case F is positively elliptic. Moreover,since π even ( F ) ⊗ Q ≤ dim π even ( M ) ⊗ Q , it follows that H ∗ ( F ) is generatedby one element. Since the Halperin conjecture is confirmed for at most 3cohomology algebra generators (see Section 1.1), the result follows againfrom Theorem D.In the second case, χ π ( F ) = 1, the space B is positively elliptic, and weshall have to distinguish yet two more non-trivial cases (using Lemma 4.2again). For this we first note that dim π even ( F ) ⊗ Q ≤ 1, dim π odd ( F ) ⊗ Q ≤ π odd ( B ) ⊗ Q ≤ 2, dim π odd ( B ) ⊗ Q ≤ 2. Combining these pieces ofinformation leads to the following cases.(i) F either rationally is an odd-dimensional sphere and dim π even ( B ) ⊗ Q = dim π odd ( B ) = 1, or dim π even ( B ) ⊗ Q = dim π odd ( B ) = 2, or(ii) F is of the type ( ∗ ) described afore the proof. Hence H ∗ ( B ) is eithergenerated by one element, by two elements or contractible.In order to apply Propositions 5.3 and 5.4 it remains to observe using Lemma5.7 that in any case all of X , F , and B are formal. 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