Relative Gottlieb groups of mapping spaces and their rational cohomology

Let f : X Ñ Y be a map of simply connected CW-complexes of finite type. Put maxπ ̊(Y )bQ = max ␣ i | πi(Y )bQ ‰ 0 ( . In this paper we compute the relative Gottlieb groups of f when X is an F0-space and Y is a product of odd spheres. Also, under reasonable hypothesis, we determine these groups when X is a product of odd spheres and Y is an F0-space. As a consequence, we show that the rationalized G-sequence associated to f splits into a short exact sequence. Finally, we prove that the rational cohomology of map(X,Y ; f) is infinite dimensional whenever maxπ ̊(Y ) b Q is even. Анотація. Нехай f : X Ñ Y – відображення однозв’язних CW-комплексів скінченного типу. Покладемо maxπ ̊(Y )bQ = max ␣ i | πi(Y )bQ ‰ 0 ( . В роботі обчислено відносні групи Готліба відображення f для випадку, коли X є F0-простом, а Y – добутком сфер непарних розмірностей. За досить природних припущень, ці групи також обчислено для протилежної ситуації, коли X – це добуток сфер непарних розмірностей, а Y – F0-простір. Як наслідок, показано, що раціоналізована G-послідовність, пов’язана з f , розщеплюється в коротку точну послідовність. Також доведено, що раціональні когомології map(X,Y ; f) мають нескінченну розмірність, за умови, що maxπ ̊(Y ) b Q є парним. 2010 Mathematics Subject Classification: Primary 55P62; Secondary 54C35.


INTRODUCTION
In this paper all spaces are simply connected CW-complex and are of finite type over Q, i.e., have finite dimensional rational cohomology in each degree.
For two topological spaces X and Y , let map(X, Y ) be the mapping space of all free continuous maps of X into Y . This space is generally disconnected with path-components corresponding to the set of free homotopy classes of maps from X to Y . We write map(X, Y ; f ) for the path-component containing a given map f : X Ñ Y .
The n-th generalized Gottlieb group G n (Y, X; f ) of a map f : X Ñ Y is the subgroup of π n (Y ) consisting of homotopy classes of map α : S n Ñ Y such that the wedge (α _ f ) : S n _ X Ñ Y extends to a map H : S nˆX Ñ Y . Alternately, G n (Y, X; f ) is the image of the map induced on homotopy groups by the evaluation map ω : map(X, Y ; f ) Ñ Y , [15]. The n-th Gottlieb group of X, denoted G n (X), is a special case X = Y and f = id X , see [4,5]. In particular, G n (X) = im ( ω 7 : π n (map(X, X; id X )) Ñ π n (X) ) .
Gottlieb groups have led to many interesting results in topology, especially, in fibration, in fixed point theory and in the theory of identification of spaces. The existence of cross-section can also be studied using the Gottlieb groups. For instance, the triviality of the n-th Gottlieb group of a space ensures that every fibration over an (n + 1)-dimensional sphere with the space as a fiber has a cross-section, [5]. But, unfortunately there are not many explicit computations of these groups in literature. One reason is the fact that a map of spaces does not necessarily induce a corresponding homomorphism of Gottlieb groups. In [16], Lee and Woo attempted to circumvent this problem by introducing the n-th relative Gottlieb group G rel n (Y, X; f ), and showed that they fit in a sequencë¨¨Ñ which is not necessarily exact. The computation of the rational relative Gottlieb groups was limited to sporadic cases, see [2,7,9,11]. Our goal in this paper is to compute these groups in some new cases. Our main results are the following: Our last result focused on rational cohomology of map(X, Y ; f ).
The paper is organized as follows. In section 2 and 3, we introduce our notation and recall some background of rational homotopy theory, namely Sullivan minimal models, derivations and mapping cone. In section 4, we use this background to prove Theorem 1.1. Section 5 is devoted to establish Theorem 1.2. In section 6 we prove that the rational cohomology of mapping spaces is infinite dimensional, in particular we prove Theorem 1.3. The paper ends with section 7 in which we propose an open problem.

SULLIVAN MINIMAL MODELS
We will work with Q as ground field and our principal tools are Sullivan minimal models. A detailed description of these and the standard tools of rational homotopy theory can be found in [1]. For our purposes, we recall the following.

Definition 2.1. A commutative differential graded algebra (cdga) is a graded algebra
for all x P A i and y P A j .
In [14], D. Sullivan defined a contravariant functor A PL which associates to each space X a cgda A PL (X) which represents the rational homotopy type of X. He also constructed, for each simply connected cgda (A, d) (i.e., satisfying H 0 (A, d) = H 1 (A, d) = 0), another cgda (ΛV, d) and a map which induces an isomorphism in cohomology, where ΛV denotes the free commutative graded algebra on the graded vector space V = ' n V n , which has a well ordered, homogeneous basis tx α u such that, if V ăα denotes span tx β | β ă αu, we have dx α P Λ ě2 (V ăα ). The cgda (ΛV, d) is called a Sullivan Next, we focus on elliptic spaces. A simply connected CW-complex X is elliptic if and only if π˚(X) b Q and H˚(X; Q) are both finite dimensional. Algebraically, this means that V and H˚(ΛV, d) are both finite dimensional, where (ΛV, d) is a Sullivan minimal model of X. There is a remarkable subclass of elliptic spaces called F 0 -space.

Definition 2.2. An elliptic space X is said to be an
Therefore, we remark that in terms of Sullivan minimal model, an F 0space X has a model of the form

DERIVATION OF A SULLIVAN MODEL AND THE RATIONALIZED G-SEQUENCE
Our purpose in this section is to give a description in rational homotopy theory of all the terms involved in the G-sequence (1.1).
Define a φ-derivation θ of degree n to be a linear map θ : ΛW Ñ ΛV that reduces degree by n such that Let Der n (ΛW, ΛV ; φ) denote the vector space of φ-derivations of degree n for n ą 0. Define a linear map Here, we note that (Der˚(ΛW, ΛV ; φ), B) is a chain complex, where In the case ΛW -ΛV and φ = id ΛV , the chain complex of derivations Der˚(ΛV, ΛV ; id ΛV ) is just the usual complex of derivations on the cdga ΛV which we denoted by Der˚(ΛV ). There is an isomorphism of graded vector spaces Der˚(ΛW, ΛV ; φ) -Hom˚(W, ΛV ). Hence, we can denote by (y, x) the unique φ-derivation sending an element y P W to x P ΛV and the other generators to zero.

Definition 3.2.
For n ě 2, the n-th generalized Gottlieb group of φ is defined as follows: Note that w˚P Hom n (W, Q), (w˚is the dual of the basis element w of W n ) is in G n (ΛW, ΛV ; φ) if and only if w˚extends to a derivation θ of Der n (ΛW, ΛV ; φ) such that B(θ) = 0. In particular, we have Definition 3.3. The n-th Gottlieb group of (ΛV, d V ) is defined as follows: Theorem 3.4 (cf. [7]). If X is a finite CW-complex, then We recall the definition of the mapping cone of a chain map φ : A Ñ B.
be a map of differential graded vector spaces. The mapping cone of φ denoted by Rel˚(φ) is defined as follows: ) .
Further, define the inclusion and projection maps J : B n Ñ Rel n (φ) by J(b) = (0, b) and P : Rel n (φ) Ñ A n´1 by P (a, b) = a. These yields a short exact sequence of chain complexes

This definition can be applied to the Sullivan minimal model
Note that the pre-composition with φ gives a map of chain complexes: φ˚: Der˚(ΛV ) Ñ Der˚(ΛW, ΛV ; φ).
Following G. Lupton and S. B. Smith [7], we consider the commutative diagram Here, ε denotes the augmentation of either ΛV or ΛW . On passing to homology and using the naturality of the mapping cone construction, we obtain the following homology ladder for n ě 2:¨¨/ Then, the rationalized G-sequence of the map φ : which ends in G 2 (ΛW, ΛV ; φ). We finish this section by an overriding hypothesis. In general, we assume that all spaces appearing in the sequel are rational simply connected CWcomplex and are of finite type. In this section, let f : X Ñ Y be a map where X is an F 0 -space and Y is products of odd dimensional spheres. We give a partial generalization of a result of G. Lupton and S. B. Smith (cf. [7,Theorem 4.3]). We begin our motivation by the following: which rationally is given by where subscripts denote degrees. The differential is given by Hence, from degree reasons and d˝φ = 0, we write φ(y 2i+1 ) = φ(y 2j+1 ) = 0. An easy argument shows that It follows that the G-sequence of f is exact. Though, if i = 1 and j = 3 we note that the homomorphism induces on rational homotopy groups f 7 is non null.
Thus, we define Now, consider the map which is null from (4.1). So, in a similar fashion as above, we get for 1 ď k ď p and 1 ď i ď m that Therefore, it is easy to see that ( { (y k , 1), 0) and (0, { (z i , 1)) are not boundaries in Rel˚( x φ˚). Combining all the above, we obtain that In summary, we have computed that The G-sequence associated to φ splits into short exact sequence Proof. Wee see directly that G˚(ΛV ) -V odd and G˚(ΛW, ΛV ; φ) -W . Therefore, to finish this proof we use Theorem 4.3 and the rationalized G-sequence (3.1). □

RELATIVE GOTTLIEB GROUPS OF MAPPING SPACES BETWEEN PRODUCTS OF ODD SPHERES AND F 0 -SPACE
Given a simply connected space X which is finite dimensional rational homotopy groups, i.e., dim π˚(X) ă 8, write and also denote by, Here (ΛV, d) denotes the Sullivan minimal model of X. Now, let f : X Ñ Y be a map such that X » Π j S 2n j +1 and Y is an F 0 -space. In this section, we use Sullivan minimal models to compute the rational relative Gottlieb groups of f . As a consequence, we show that the G-sequence of the map f splits into a short exact sequence. Theorem 5.1. Given a map f : Π j S 2n j +1 Ñ Y and φ : (ΛW, d) Ñ (ΛV, 0) its Sullivan minimal model. If W odd -V and max W even ď min W odd , then: (1) G˚(ΛW, ΛV ; φ) -W ; (2) G rel (ΛW, ΛV ; φ) -W even .
Proof. We give the proof by dividing it into two cases. where |x i | is even. The differential is defined as follows: Under the condition max W even ď min W odd , the Sullivan minimal model of f , φ : (Λ(x 1 , . . . , x m , y 1 , . . . , y m ), d) Ñ (Λ(y 1 , . . . , y m ), 0) is given on generators by where α P Λ(y 1 , . . . , y m ); α may be null. Now, an easy computation reveals that each of the derivations (y j , 1) is a cycle and cannot be boundary for 1 ď j ď m. We deduce that [(y j , 1)] is non null in H˚(Der(ΛW, ΛV ; φ)).

RATIONAL COHOMOLOGY OF MAPPING SPACES
Let f : X Ñ Y be a map between simply connected CW-complexes.
is the path components of the space of maps X Ñ Y consisting of those maps that are homotopic to f . The first description of a Sullivan model, in particular of mapping spaces is due to A. Haefliger [6]. In [7], G. Lupton and S. B. Smith gave an elegant formula for the rational homotopy type of map(X, Y ; f ) in terms of Sullivan minimal models of X and Y . In particular, they proved that when X is finite π i (map(X, Y ; f )) -H i (Der(ΛW, ΛV ; φ)) for i ě 2. where G(k, n) is the complex Grassmannians, [10]. However, there is no explicit and complete description of the rational cohomology of map(X, Y ; f ). We begin by the following: From (6.1), it follows that π i (map(X, Y ; f )) -Q for i = 2m, 2m´2n´1 and zero otherwise.
Up to this point, our observation is well-known and also easily obtained by a number of standard methods. We can hence generalize this example as follows: Theorem 6.2. Let f : X Ñ Y be a map with X is finite and max π˚(Y ) is even. Then map(X, Y ; f ) has infinite dimensional rational cohomology.