On a Conjecture of Yamaguchi and Yokura

Let F⟶ E⟶ p B 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∗(p)Q ≤ dim Ker H∗(p;Q) + 1. Our goal is to prove this conjecture when F and B satisfy the condition πeven(F)Q � πeven(B)Q � 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.


Introduction
We begin with a description of the conjecture referred to in the title. In this paper, all spaces are simply connected CWcomplexes and are of finite type, i.e., have finite dimensional rational cohomology.
A space X is said to be elliptic if the dimensions of cohomology and homotopy are both finite [1]. For these spaces, Hilali [2] conjectured in 1990. (Hilali). Let X be a simply connected rationally elliptic space; then,

Conjecture 1
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].
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]. e rational homotopy type of X is encoded in a differential graded algebra (A, d) called the Sullivan minimal model of X. is is a free-graded algebra A � ΛV generated by a graded vector space V � ⊕ i≥2 V i and with decomposable differential, i.e., Notice that (ΛV, d) determines the rational homotopy type of X. Especially, there are isomorphisms: H * (ΛV, d) � H * (X; Q) as graded commutative algebras, V � π * (X) Q as graded vector spaces.

(5)
Although our results are stated and proved in purely algebraic terms, they do admit topological interpretations via this correspondence. erefore, we can also characterize an elliptic space in terms of its Sullivan minimal model. A space X 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.
Let F ⟶ E ⟶ p B be a fibration. e KS-model for p is a short exact sequence of DGA, with (ΛW, d) and (ΛV, d F ) as the Sullivan minimal models for B and F, respectively (see [1], Proposition 15.5).
e 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.
is the KS-model for a rational fibration of elliptic spaces, then where D 1 is the linear part of D.
is 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 fibration F ⟶ i E ⟶ B is totally non- It is equivalent to requiring that the Serre spectral sequence collapses at E 2 -term. In this case, there is an isomorphism: We end this section with some notations and conventions. In general, we use V or W to denote a positively graded rational vector space of the finite type. e cohomology of a DGA (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.

The Conjecture of Yamaguchi and Yokura
e topological aspect in this section is centered around the following question.
be the KS-model of a fibration with W even � V even � 0, is it true that dim Coker H * (J, D 1 ) ≤ dim Coker H * (J) + 1? Our most general results here are as follows.
(1) F has the rational homotopy type of a product of odddimensional spheres (2) p admits a section en, the conjecture (YY) is true.
Recall that a fibration F ⟶ E ⟶ p B admits a section if there is a map s: B ⟶ E such that pos≃Id B . However, in [11], Lemma 3, omas showed that a fibration admits a section if and only if there exists a KS-model: Proof of eorem 1. In the following, we make the iden- e second assumption implies that Dv 1 � 0 and is allows us to deduce that D is decomposable, and then On the contrary, we put Furthermore, a direct argument shows that every element in Γ is a nonexact D-cycle. Since the elements of Γ all have different degrees, then they are linearly independent. erefore, from (11) we have □ Example 1. Let us consider the fibration given by the KS-model where |x| � 2, |y| � 7, |u| � 3, and |v| � 5, and the nonzero differentials are given by: Dy � x 4 and Dv � x 3 . Hence, we We can see that p does not admit a section, though it satisfies the conjecture (YY).
e first hypothesis implies that the Sullivan minimal model (ΛV, d F ) for F is oddly generated, i.e., V even � 0, and then we write V � 〈v 1 , v 2 , . . . , v n 〉 with |v i | ≤ |v j | whenever i < j and each |v i | are odd. From the second hypothesis, the Sullivan minimal model (ΛW, d) for B has trivial differential, d � 0, with W even � 0; more precisely, we denote (Λ(u 1 , u 2 , . . . , u m ), 0) with |u j | is odd and m ≥ n − 2.
erefore, the KS-model of p is given by en, for degree reasons, D is decomposable. Hence, we clearly have dim Coker H * J, D 1 � n. (18) In order to prove it suffices to find at least n elements in H * (Λ(u 1 , . . . , u m ) ⊗ Λ(v 1 , . . . , v n ), D)/Im H * (J). For this, we put (20) Here, the notation u i means that the element u i is removed, and μ E and μ F 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. us, □ Example 2. Note that condition (30) above is sufficient but not necessary. Indeed, consider the nontrivial fibration F ⟶ E ⟶ S 3 × S 5 given by the following KS-model:  (Λ(x, y, z, t, u, v, w), D), we deduce that dim Coker H * (J) + 1 ≥ 5.
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.
As DW � 0, then every element in ΛW is a D-cycle and since DV � Λ 2 W; thus, the cohomology represented by elements of word-length at least two is bounded. is proves that taking into account zeroth cohomology, and then we obtain e abovementioned computation works for n ≠ 2. If n � 2, referring to the Sullivan minimal model in this case, we have (ΛW ⊗ ΛV, D) � (Λ(x, y, z), D) with the degrees of all elements are odd and nonzero differential Dz � xy. It is easy to check that dim Coker H * (J) + 1 � 4 ≥ 1 � dim Coker H * J, D 1 .

The Hilali Conjecture
In this section, we consider a particular question suggested by Hilali and Mamouni in [6].

Remark 1.
is result is also true if e � 0.
□ Proposition 4. Let F ⟶ E ⟶ B be a TNCZ fibration in which F and B satisfy the conjecture (H), and then E will too.

Proof.
By assumption, we have H * (E; Q) � H * (F; Q) ⊗ H * (B; Q). Next, we argue exactly as in the Proof of eorem 3 to show that dim H * (E; Q) ≥ dim π * (E) Q .
A much stronger consequence follows if we restrict the fibre. □ Corollary 1. Let F ⟶ E ⟶ B be a fibration, in which F is an F 0 -space with rank π odd (F) Q ≤ 3. en, E satisfies the conjecture (H) once B satisfies it.
Proof. It is an immediate consequence from [14]. □ Proposition 5. Let F ⟶ E ⟶ B be a fibration such that π even (F) Q � π even (B) Q � 0, and then E satisfies the conjecture (H). e proof of this proposition is omitted. It can be proved using the result of the paper [9].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.