Evaluation subgroups of a map and the rationalized G -sequence

. In this paper, we determine, in terms of the Sullivan models, rational evaluation subgroups of the inclusion C P ( n ) (cid:44) → C P ( n + k ) between complex projective spaces and, more generally, the G -sequence of the homotopy monomorphism ι : X (cid:44) → Y between simply connected formal homogeneous spaces for which π ∗ ( Y ) ⊗ Q is ﬁnite dimensional.


Introduction
Let us remind the notion of a Gottlieb group (see [3]). Given a based CWcomplex X, an element α ∈ π n (X) is a Gottlieb element of X if (α, id X ) : X ∨ S n → X extends toα : X × S n → X. The set G n (X) of all Gottlieb elements α ∈ π n (X) is called the n-th Gottlieb group of X or the n-th evaluation subgroup of π n (X).
Gottlieb groups play a profound role in topology, covering spaces, fixed point theory, homotopy theory of fibrations, and other fields. For instance, the triviality of Gottlieb groups is related to the cross section problem of fibrations.
Further, let f : X → Y be a based map of simply connected finite CWcomplexes. As it was shown in [4], the evaluation at the basepoint of X gives the evaluation map ω : Map(X, Y ; f ) → Y , where Map(X, Y ; f ) is the component of f in the space of mappings from X to Y . The image of the homomorphism induced in homotopy groups ω : π * Map(X, Y ; f ) → π * (Y ) is called the n-th evaluation subgroup of p and it is denoted by G n (Y, X; p).
Note that if f = id X , the space Map(X, Y ; f ) is the monoid aut 1 (X) of selfequivalences of X homotopic to the identity of X, then ev : aut 1 (X) → X is the evaluation map, and the image of the induced homomorphism ev : π * (aut 1 (X)) → π * (X) is G n (X), i.e., the n-th Gottlieb group.
In [10], Woo and Lee studied the relative evaluation subgroups G rel n (X, Y ; p) and proved that they fit in a sequence called the G-sequence of f . This sequence is exact in some cases, for instance, if f is a homotopy monomorphism.
Recently, Smith and Lupton [4] identified the homomorphism induced on rational homotopy groups by the evaluation map ω : Map(X, Y ; f ) → Y in terms of a map of complexes of derivations constructed directly from the Sullivan minimal model of f . In [5,6], rationalized evaluation subgroups of mapping spaces between Complex Grassmann manifolds G k,n (C), which are a generalization of complex projective spaces, were studied.
In this paper, we use a map of complexes of derivations of minimal Sullivan models of mapping spaces to compute rational relative Gottlieb groups of the inclusion CP (n) → CP (n + k). More generally, we consider the inclusion ι : X → Y between simply connected formal homogeneous spaces for which π * (Y ) ⊗ Q is finite dimensional.

Preliminaries
Through this paper, we rely on the theory of minimal Sullivan models in rational homotopy theory for which [1] is our standard reference. All vector spaces and algebras are taken over a field Q of rational numbers. We start with recalling some definitions. Definition 1.1 A commutative graded differential algebra (cdga) is a graded algebra (A, d) such that xy = (−1) |x||y| yx and d(xy then ∧V denotes the free commutative graded algebra defined by the tensor product where S(V even ) is the symmetric algebra on V even and E(V odd ) is the exterior algebra on V odd . Definition 1.2 A Sullivan algebra is a commutative differential graded algebra (∧V, d) where V = ∪ k≥0 V (k) and V (0) ⊂ V (1) · · · such that dV (0) = 0 and dV (k) ⊂ ∧V (k − 1). It is called minimal if dV ⊂ ∧ ≥2 V .
If (A, d) is a cdga of which the cohomology is connected and finite dimensional in each degree, then there always exists a quasi-isomorphism from a Sullivan algebra (∧V, d) to (A, d) [1]. To each simply connected space, Sullivan associates a cdga A P L (X) of rational polynomial differential forms on X that uniquely determines the rational homotopy type of X [8]. A minimal Sullivan model of X is a minimal Sullivan model of A P L (X). More precisely, H * (∧V, d) ∼ = H * (X; Q) as graded algebras and V ∼ = π * (X) ⊗ Q as graded vector spaces.
Let (A, d) be a cdga. A derivation θ of degree k is a linear mapping θ : A n → A n−k such that θ(ab) = θ(a)b + (−1) k|a| aθ(b). Denote by Der k A the vector space of all derivations of degree k and Der A = ⊕ k Der k A. The commutator bracket induces a graded Lie algebra structure on Der A. Moreover, (Der A, δ) is a differential graded Lie algebra (see, for example, [8]), with the differential δ defined in the usual way by The differential graded vector space of φ-derivations is denoted by (Der(A, B; φ), ∂), where the differential ∂ is defined by In the case A = B and φ = 1 B , the vector space (Der(B, B; 1), ∂) is just a usual differential graded Lie algebra of derivations on the cdga B (see [4]). Note that, there is an isomorphism of graded vector spaces It was shown in [4], that a pre-composition with φ gives a chain complex map φ * : Der(B, B; 1) → Der(A, B; φ), and a post-composition with the augmentation ε : B → Q gives a chain complex map ε * : Der(A, B; φ) → Der(A, Q; ε). The evaluation subgroup of φ is defined as follows G n (A, B; φ) = Im{H(ε * ) : H n (Der(A, B; φ)) → H n (Der(A, Q; ε))}.
In particular, G n (B) ∼ = G n (X Q ) if B is the minimal Sullivan model of a simply connected space X [1, Proposition 29.8].
Examples of formal spaces include spheres, projective complex spaces, homogeneous spaces G/H where G and H have same rank, and compact Kähler manifolds.

Evaluation subgroups of a map and the Gsequence
Consider the inclusion CP (n) → CP (n + k). In [2], the minimal model of CP (n) is given by (∧( , and the minimal model of CP (n + k) is given by (∧y 2 , y 2(n+k)+1 ), d) with dy 2 = 0, dy 2(n+k)+1 = y n+k+1 2 . Moreover, the inclusion CP (n) → CP (n + k) has a model of the form 0) and dP = 0, dY ⊆ ∧P . The associated minimal Sullivan model (∧V, d) is called a pure Sullivan algebra. Homogeneous spaces are pure. Further, since X is a formal homogeneous space, we have where (α 1 , . . . , α k ) is a regular sequence in ∧P . Hence, X as a formal homogeneous space admitting a minimal Sullivan model of the form (∧V, d) = (∧(P ⊕ Y ), d) ⊗ (∧W, 0), where dP = 0, dy k = α k . We study the evaluation subgroups of φ and, more generally, the inclusion ι between formal spaces. We have the following results.
From the theorem above, it follows that the G-sequence reduces to and it is exact.
It is easy to see that δθ t (y t ) = 0. Hence, the generators y t , t ∈ {1, . . . , k} are Gottlieb elements. It follows in the same way that the generators w 1 , . . . , w r are Gottlieb elements.
Theorem 2.6 Let X be a simply connected formal homogeneous spaces for which π * (X) ⊗ Q is finite dimensional, and let B = (∧(V 0 ⊕ V 1 ), d) be its minimal Sullivan model.