# On a Conjecture of Yamaguchi and Yokura

Authors: Abdelhadi Zaim; Saloua Chouingou; Mohamed Anas Hilali
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

LetF⟶E⟶pB be a fibration of simply connected elliptic spaces. Our paper investigates the conjecture proposed by T. Yamaguchi and S. Yokura, states that dim Ker π∗pℚ≤ dim Ker H∗p;ℚ+1. Our goal is to prove this conjecture when F and B satisfy the condition πevenFℚ=πevenBℚ=0. We go also on to establish a well-known conjecture of Hilali for a class of spaces which puts it into the context of fibration.


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## Body

## 1. Introduction

We begin with a description of the conjecture referred to in the title. In this paper, all spaces are simply connected CW-complexes and are of finite type, i.e., have finite dimensional rational cohomology.A spaceX is said to be elliptic if the dimensions of cohomology and homotopy are both finite [1]. For these spaces, Hilali [2] conjectured in 1990.Conjecture 1 (Hilali).
LetX be a simply connected rationally elliptic space; then,(1)Hdimπ∗Xℚ≤dimH∗X;ℚ.Generally, speaking about cohomology is delicate, invariant, and difficult to compute. Recently, Yamaguchi and Yokura proposed another version to the conjecture (H) of a map [3].Conjecture 2 (Yamaguchi–Yokura).
Letp:E⟶B be a continuous map between two elliptic spaces, then(2)YYdim Kerπ∗pℚ≤dim KerH∗p;ℚ+1,where(3)Kerπ∗pℚ≔⊕i≥1Kerπipℚ:πiEℚ⟶πiBℚ,KerH∗p;ℚ≔⊕i≥0KerHip;ℚ:HiE;ℚ⟶HiB;ℚ.In particular, ifB≃∗, we obtain the conjecture (H). Then, a positive answer to the conjecture (YY) would give a positive answer to the conjecture (H).Although we will recall some basic facts about Sullivan minimal models, our proofs assume a working familiarity with them. Our reference for rational homotopy theory is [1]. The rational homotopy type of X is encoded in a differential graded algebra A,d called the Sullivan minimal model of X. This is a free-graded algebra A=ΛV generated by a graded vector space V=⊕i≥2Vi and with decomposable differential, i.e.,(4)d:Vi⟶Λ≥2Vi+1.Notice thatΛV,d determines the rational homotopy type of X. Especially, there are isomorphisms:(5)H∗ΛV,d≅H∗X;ℚas graded commutative algebras,V≅π∗Xℚas graded vector spaces.Although our results are stated and proved in purely algebraic terms, they do admit topological interpretations via this correspondence. Therefore, we can also characterize an elliptic space in terms of its Sullivan minimal model. A spaceX with Sullivan minimal model ΛV,d is elliptic if V and H∗ΛV,d are both finite dimensional. Also, we can reformulate the conjecture (H) algebraically as follows.Conjecture 3.
IfΛV,d is a simply connected elliptic Sullivan minimal model, then(6)dimV≤dimH∗ΛV,dH.This conjecture is open in general, but has been proved in some interesting cases (see [2, 4–9]).LetF⟶E⟶pB be a fibration. The KS-model for p is a short exact sequence(7)ΛW,d⟶ΛW⊗ΛV,D⟶ΛV,dF,of DGA, with ΛW,d and ΛV,dF as the Sullivan minimal models for B and F, respectively (see [1], Proposition 15.5). The differential D satisfies Dw=dw for w∈W and Dv−dFv∈Λ+W⊗ΛV for v∈V. The DGA ΛW⊗ΛV,D is a Sullivan model for the total space E but is not, in general, minimal.In view of the notation above, the algebraic version of the conjecture (YY) is given.Conjecture 4.
IfΛW,d⟶JΛW⊗ΛV,D⟶ΛV,dF is the KS-model for a rational fibration of elliptic spaces, then(8)YYdim CokerH∗J,D1≤dim CokerH∗J+1,where D1 is the linear part of D.This conjecture is affirmed for spherical fibration and TNCZ fibration whose fibre satisfies the conjecture (H) (see [3]). In a previous joined work, the authors with Hilali have shown this conjecture for fibrations whose fibre has at most two oddly generators and also in the case of H⟶G⟶G/H, where G is a compact connected Lie group and H is a closed subgroup of G (see [10]).Recall that a fibrationF⟶iE⟶B is totally noncohomologous to zero (abbreviated TNCZ), if the induced homomorphism H∗i:H∗E;ℚ⟶H∗F;ℚ is surjective. It is equivalent to requiring that the Serre spectral sequence collapses at E2-term. In this case, there is an isomorphism: H∗E;ℚ≅H∗F;ℚ⊗H∗B;ℚ of H∗B;ℚ-modules.We end this section with some notations and conventions. In general, we useV or W to denote a positively graded rational vector space of the finite type. The cohomology of a DGA A,d is denoted H∗A,d or just H∗A, and let x∈H∗A,d stand for the cohomology class of the cocycle x∈A.As an overriding hypothesis, we assume that all spaces appearing in this paper are rational simply connected elliptic spaces.


## 2. The Conjecture of Yamaguchi and Yokura

The topological aspect in this section is centered around the following question.Question 1.
LetΛW,d⟶JΛW⊗ΛV,D⟶ΛV,dF be the KS-model of a fibration with Weven=Veven=0, is it true that dim Coker H∗J,D1≤ dim Coker H∗J+1?
Our most general results here are as follows.Theorem 1.
LetF⟶E⟶pB be a fibration. Suppose(1)
F has the rational homotopy type of a product of odd-dimensional spheres(2)
p admits a section
Then, the conjecture (YY) is true.Recall that a fibrationF⟶E⟶pB admits a section if there is a map s:B⟶E such that pos≃IdB. However, in [11], Lemma 3, Thomas showed that a fibration admits a section if and only if there exists a KS-model:(9)ΛW,d⟶ΛW⊗ΛV,D⟶ΛV,dF,such that Dv−dFv∈Λ+W⊗Λ+V for v∈V.Proof of Theorem 1.
In the following, we make the identificationWn≅πnBℚ, where ΛW,d is the Sullivan minimal model of B. Hypothesis (11) implies that the Sullivan minimal model ΛV,dF for F has trivial differential, dF=0, with Veven=0. Write V=v1,v2,…,vn with vi≤vj whenever i<j and each vi is odd. So, the KS-model for F⟶E⟶pB is of the form(10)ΛW,d⟶JΛW⊗Λv1,v2,…,vn,D⟶Λv1,v2,…,vn,0.
The second assumption implies thatDv1=0 and Dvi∈Λ+W⊗Λ+v1,v2,…,vi−1 for 2≤i≤n. This allows us to deduce that D is decomposable, and then(11)dim CokerH∗J,D1=dimV=n.
On the contrary, we put(12)Γ=v1,v1v2,v1v2v3,…,v1v2v3…vifor4≤i≤n.
Furthermore, a direct argument shows that every element inΓ is a nonexact D-cycle. Since the elements of Γ all have different degrees, then they are linearly independent. Therefore, from (11) we have(13)dim CokerH∗J+1≥n+1≥dim CokerH∗J,D1.Example 1.
Let us consider the fibration(14)S3×S5⟶X⟶pℂℙ3,given by the KS-model(15)Λx,y,d⟶JΛx,y⊗Λu,v,D⟶Λu,v,dF,where x=2, y=7, u=3, and v=5, and the nonzero differentials are given by: Dy=x4 and Dv=x3. Hence, we have dim Coker H∗J,D1=2 and Coker H∗J=ℚu,ux,ux2,y−xv. Then,(16)1+dim CokerH∗J=5≥dim CokerH∗J,D1.
We can see thatp does not admit a section, though it satisfies the conjecture (YY).Theorem 2.
LetF⟶E⟶pB be a fibration. Suppose(1)
πevenFℚ=0(2)
B has the rational homotopy type of product of at least dim π∗Fℚ−2 odd-dimensional spheres.
Then, the conjecture (YY) holds.Proof.
According to the dimension ofπ∗Fℚ, we distinguish two cases:Case I: dimπ∗Fℚ≤2, so B has the rational homotopy type of a point. This implies that Ker π∗pℚ=π∗Fℚ and Ker H∗p;ℚ=H∗F;ℚ. From hypothesis (11) and [9], Theorem 1.2, we deduce that p satisfies the conjecture (YY).Case II: dimπ∗Fℚ≥3, let n = dim π∗Fℚ and m= dim π∗Bℚ.
The first hypothesis implies that the Sullivan minimal modelΛV,dF for F is oddly generated, i.e., Veven=0, and then we write V=v1,v2,…,vn with vi≤vj whenever i<j and each vi are odd. From the second hypothesis, the Sullivan minimal model ΛW,d for B has trivial differential, d=0, with Weven=0; more precisely, we denote Λu1,u2,…,um,0 with uj is odd and m≥n−2. Therefore, the KS-model of p is given by(17)Λu1,…,um,0⟶JΛu1,…,um⊗Λv1,…,vn,D⟶Λv1,…,vn,dF.
Then, for degree reasons,D is decomposable. Hence, we clearly have(18)dim CokerH∗J,D1=n.
In order to prove(19)dim CokerH∗J+1≥dim CokerH∗J,D1,it suffices to find at least n elements in H∗Λu1,…,um⊗Λv1,…,vn,D/ImH∗J. For this, we put(20)Γ=μE,u1u2,…,u^i,…,umμFfor1≤i≤m.
Here, the notationu^i means that the element ui is removed, and μE and μF denote the fundamental class of E and F, respectively. Now, for each element α in Γ, we have Dα=0, and it is easy to see that α cannot be a D -coboundary. Thus, α is in Coker H∗J for each α in Γ. Consequently, we have(21)dim CokerH∗J+1≥m+2≥n.Example 2.
Note that condition (30) above is sufficient but not necessary. Indeed, consider the nontrivial fibration F⟶E⟶S3×S5 given by the following KS-model:(22)Λx,y,0⟶JΛx,y⊗Λz,t,u,v,w,D⟶Λz,t,u,v,w,dF,with x=y=z=t=3, u=v=5, w=7, Dx=Dy=Dz=Dt=0, Du=xy+zt, Dv=zt, and Dw=zv. A careful check reveals that D defines a differential. Since Im H∗J=ℚ1,x,y,xy and ℚz,t,tv,zw⊂H∗Λx,y,z,t,u,v,w,D, we deduce that(23)dim CokerH∗J+1≥5.
Thus, the conjecture (YY) is true though dim π∗Bℚ< dim π∗Fℚ−2.
In the remainder of this section, we show the conjecture (YY) for certain fibrations whose total space has a two-stage Sullivan minimal model ΛU,D, i.e., U decomposes as U≅W⊕V with dW=0 and dV⊂ΛW. Furthermore, if ΛU,D is elliptic, then W may have generators of odd or even degree but V must have generators of odd degree only.Proposition 1.
LetΛW⊗ΛV,D be a two-stage Sullivan minimal model with Weven and V have the same dimensional, then the following KS-extension(24)ΛWodd,0⟶JΛW⊗ΛV,D⟶ΛWeven⊗ΛV,dF,satisfies the conjecture (YY).Proof.
The conditiondimWeven=dimVimplies thatΛWeven⊗ΛV,dF is the Sullivan minimal model of an F0-space ([1], Section 32). Moreover, since DWeven=0 and DVodd⊂ΛWodd⊗ΛWeven, because ΛW⊗ΛV,D is supposed two-stage Sullivan minimal model, then J is a pure fibration and by Theorem 2 of [11], it is TNCZ. Furthermore, from [3], we deduce the result.Proposition 2.
LetΛW⊗ΛV,D be a two-stage Sullivan minimal model with odd degree only and assume that D:V⟶Λ2W is an isomorphism, then the following KS-extension(25)ΛW,0⟶JΛW⊗ΛV,D⟶ΛV,dF,satisfies the conjecture (YY).Proof.
Suppose that dimW=n, since D:V⟶Λ2W is an isomorphism, and ΛW is an exterior algebra, we have dim V=n2−n/2. On the contrary, we know from Proposition 2.1 of [12](26)dimH∗ΛW⊗ΛV,D≥2n2−n/2.
AsDW=0, then every element in ΛW is a D-cycle and since DV≅Λ2W; thus, the cohomology represented by elements of word-length at least two is bounded. This proves that(27)dim ImH∗J=dimW+1=n+1,taking into account zeroth cohomology, and then we obtain (28)dim CokerH∗J+1≥2n2−n/2−n+1+1≥n2−n2.
The abovementioned computation works forn≠2. If n=2, referring to the Sullivan minimal model in this case, we have ΛW⊗ΛV,D=Λx,y,z,D with the degrees of all elements are odd and nonzero differential Dz=xy. It is easy to check that(29)dim CokerH∗J+1=4≥1=dim CokerH∗J,D1.


## 3. The Hilali Conjecture

In this section, we consider a particular question suggested by Hilali and Mamouni in [6].Question 2.
IfF⟶E⟶B is a fibration where F and B are elliptic spaces and both satisfy the conjecture (H), is it true that E will too?
As a first result concerning this question, we have the following.Theorem 3.
LetS2n+1⟶E⟶pB be a fibration in which B satisfies the conjecture (H). If p admits a section, then E satisfies the conjecture (H).A key ingredient to the proof of this theorem is as follows.Proposition 3.
LetS2n+1⟶E⟶pB be a fibration such that p admits a section. Then, we have H∗E;ℚ≅H∗S2n+1;ℚ⊗H∗B;ℚ.Proof.
SinceB is simply connected, then this fibration is oriented. Hence, we may formulate the Gysin sequence for p as follows ([13], p.375):(30)⋯⟶Hi−2n−2B;ℚ⟶δHiB;ℚ⟶p∗HiE;ℚ⟶∂Hi−2n−1B;ℚ⟶δHi+1B;ℚ⟶⋯,where δb=b⌣e for some e∈H2n+2B;ℚ. Since HiB;ℚ=0 for i<0, then the long exact sequence (30) restricts to isomorphisms:(31)0⟶HiB;ℚ⟶p∗HiE;ℚ⟶0,fori≤2n.
Fori≥2n+1, sequence (30) will be rewritten as follows:(32)0⟶H2n+1B;ℚ⟶p∗H2n+1E;ℚ⟶∂H0B;ℚ⟶δH2n+2B;ℚ⟶⋯.
Our hypothesis implies thatp∗ is injective. So, the sequence (32) breaks up into split short exact sequences:(33)0⟶HiB;ℚ⟶p∗HiE;ℚ⟶∂Hi−2n−1B;ℚ⟶0,fori≥2n+1.
From (31) and (33), we obtain(34)HiE;ℚ≅HiB;ℚ⊕Hi−2n−1B;ℚ.
On the contrary, we clearly have(35)⊕j+k=iHjS2n+1;ℚ⊗HkB;ℚ≅HiB;ℚ⊕Hi−2n−1B;ℚ,and then by the Künneth Formula, we deduce that(36)H∗E;ℚ≅H∗S2n+1;ℚ⊗H∗B;ℚ.Remark 1.
This result is also true ife=0.Proof of Theorem 3.
We have proved in Proposition3 that(37)H∗E;ℚ≅H∗S2n+1;ℚ⊗H∗B;ℚ.
So, by taking their dimension, we obtain(38)dimH∗E;ℚ≥dimH∗S2n+1;ℚ⋅dimH∗B;ℚ≥2dimπ∗BℚBsatisfies the conjectureH≥dimπ∗Eℚ,as required.Proposition 4.
LetF⟶E⟶B be a TNCZ fibration in which F and B satisfy the conjecture (H), and then E will too.Proof.
By assumption, we haveH∗E;ℚ≅H∗F;ℚ⊗H∗B;ℚ. Next, we argue exactly as in the Proof of Theorem 3 to show that dim H∗E;ℚ≥dimπ∗Eℚ.
A much stronger consequence follows if we restrict the fibre.Corollary 1.
LetF⟶E⟶B be a fibration, in which F is an F0-space with rank πoddFℚ≤3. Then, E satisfies the conjecture (H) once B satisfies it.Proof.
It is an immediate consequence from [14].Proposition 5.
LetF⟶E⟶B be a fibration such that πevenFℚ=πevenBℚ=0, and then E satisfies the conjecture (H).The proof of this proposition is omitted. It can be proved using the result of the paper [9].



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