Abstract
Let be a fibration of simply connected elliptic spaces. Our paper investigates the conjecture proposed by T. Yamaguchi and S. Yokura, states that dim Ker dim Ker . Our goal is to prove this conjecture when and satisfy the condition . 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.
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 space 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). Let be a simply connected rationally elliptic space; then,
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). Let be a continuous map between two elliptic spaces, thenwhere
In particular, if , 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 is encoded in a differential graded algebra called the Sullivan minimal model of . This is a free-graded algebra generated by a graded vector space and with decomposable differential, i.e.,
Notice that determines the rational homotopy type of . Especially, there are isomorphisms:
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 space with Sullivan minimal model is elliptic if and are both finite dimensional. Also, we can reformulate the conjecture (H) algebraically as follows.
Conjecture 3. If is a simply connected elliptic Sullivan minimal model, then
This conjecture is open in general, but has been proved in some interesting cases (see [2, 4–9]).
Let be a fibration. The KS-model for is a short exact sequenceof DGA, with and as the Sullivan minimal models for and , respectively (see [1], Proposition 15.5). The differential satisfies for and for . The DGA is a Sullivan model for the total space but is not, in general, minimal.
In view of the notation above, the algebraic version of the conjecture (YY) is given.
Conjecture 4. If is the KS-model for a rational fibration of elliptic spaces, thenwhere is the linear part of .
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 , where is a compact connected Lie group and is a closed subgroup of (see [10]).
Recall that a fibration is totally noncohomologous to zero (abbreviated TNCZ), if the induced homomorphism is surjective. It is equivalent to requiring that the Serre spectral sequence collapses at -term. In this case, there is an isomorphism: of -modules.
We end this section with some notations and conventions. In general, we use or to denote a positively graded rational vector space of the finite type. The cohomology of a DGA is denoted or just , and let stand for the cohomology class of the cocycle .
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. Let be the KS-model of a fibration with , is it true that dim Coker dim Coker
Our most general results here are as follows.
Theorem 1. Let be a fibration. Suppose(1) has the rational homotopy type of a product of odd-dimensional spheres(2) admits a sectionThen, the conjecture (YY) is true.
Recall that a fibration admits a section if there is a map such that . However, in [11], Lemma 3, Thomas showed that a fibration admits a section if and only if there exists a KS-model:such that for .
Proof of Theorem 1. In the following, we make the identification , where is the Sullivan minimal model of . Hypothesis (11) implies that the Sullivan minimal model for has trivial differential, , with . Write with whenever and each is odd. So, the KS-model for is of the formThe second assumption implies that and for . This allows us to deduce that is decomposable, and thenOn the contrary, we putFurthermore, a direct argument shows that every element in is a nonexact D-cycle. Since the elements of all have different degrees, then they are linearly independent. Therefore, from (11) we have
Example 1. Let us consider the fibrationgiven by the KS-modelwhere , , , and , and the nonzero differentials are given by: and . Hence, we have dim Coker and Coker . Then,We can see that does not admit a section, though it satisfies the conjecture (YY).
Theorem 2. Let be a fibration. Suppose(1)(2) has the rational homotopy type of product of at least dim odd-dimensional spheres.Then, the conjecture (YY) holds.
Proof. According to the dimension of , we distinguish two cases: Case I: dim , so has the rational homotopy type of a point. This implies that Ker and Ker . From hypothesis (11) and [9], Theorem 1.2, we deduce that satisfies the conjecture (YY). Case II: dim , let = dim and dim .The first hypothesis implies that the Sullivan minimal model for is oddly generated, i.e., , and then we write with whenever and each are odd. From the second hypothesis, the Sullivan minimal model for has trivial differential, , with ; more precisely, we denote with is odd and . Therefore, the KS-model of is given byThen, for degree reasons, D is decomposable. Hence, we clearly haveIn order to proveit suffices to find at least elements in . For this, we putHere, the notation means that the element is removed, and and denote the fundamental class of and , respectively. Now, for each element in , we have , and it is easy to see that cannot be a -coboundary. Thus, is in Coker for each in . Consequently, we have
Example 2. Note that condition (30) above is sufficient but not necessary. Indeed, consider the nontrivial fibration given by the following KS-model:with , , , , , , and . A careful check reveals that defines a differential. Since Im and , we deduce thatThus, the conjecture (YY) is true though dim dim .
In the remainder of this section, we show the conjecture (YY) for certain fibrations whose total space has a two-stage Sullivan minimal model , i.e., decomposes as with and . Furthermore, if is elliptic, then may have generators of odd or even degree but must have generators of odd degree only.
Proposition 1. Let be a two-stage Sullivan minimal model with and have the same dimensional, then the following KS-extensionsatisfies the conjecture (YY).
Proof. The condition is the Sullivan minimal model of an -space ([1], Section 32). Moreover, since and , because is supposed two-stage Sullivan minimal model, then is a pure fibration and by Theorem 2 of [11], it is TNCZ. Furthermore, from [3], we deduce the result.
Proposition 2. Let be a two-stage Sullivan minimal model with odd degree only and assume that is an isomorphism, then the following KS-extensionsatisfies the conjecture (YY).
Proof. Suppose that dim , since is an isomorphism, and is an exterior algebra, we have dim . On the contrary, we know from Proposition 2.1 of [12]As , then every element in is a -cycle and since ; thus, the cohomology represented by elements of word-length at least two is bounded. This proves thattaking into account zeroth cohomology, and then we obtain The abovementioned computation works for . If , referring to the Sullivan minimal model in this case, we have with the degrees of all elements are odd and nonzero differential . It is easy to check that
3. The Hilali Conjecture
In this section, we consider a particular question suggested by Hilali and Mamouni in [6].
Question 2. If is a fibration where and are elliptic spaces and both satisfy the conjecture (H), is it true that will too?
As a first result concerning this question, we have the following.
Theorem 3. Let be a fibration in which satisfies the conjecture (H). If admits a section, then satisfies the conjecture (H).
A key ingredient to the proof of this theorem is as follows.
Proposition 3. Let be a fibration such that admits a section. Then, we have .
Proof. Since is simply connected, then this fibration is oriented. Hence, we may formulate the Gysin sequence for as follows ([13], p.375):where for some . Since for , then the long exact sequence (30) restricts to isomorphisms:For , sequence (30) will be rewritten as follows:Our hypothesis implies that is injective. So, the sequence (32) breaks up into split short exact sequences:From (31) and (33), we obtainOn the contrary, we clearly haveand then by the Künneth Formula, we deduce that
Remark 1. This result is also true if .
Proof of Theorem 3. We have proved in Proposition 3 thatSo, by taking their dimension, we obtainas required.
Proposition 4. Let be a TNCZ fibration in which and satisfy the conjecture (H), and then will too.
Proof. By assumption, we have . Next, we argue exactly as in the Proof of Theorem 3 to show that dim .
A much stronger consequence follows if we restrict the fibre.
Corollary 1. Let be a fibration, in which is an -space with rank . Then, satisfies the conjecture (H) once satisfies it.
Proof. It is an immediate consequence from [14].
Proposition 5. Let be a fibration such that , and then satisfies the conjecture (H).
The 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
The authors declare that they have no conflicts of interest.