A basic object of study in homotopy theory is the Gottlieb groups. They are very interesting homotopy invariants but their calculations in general are difficult. As is well known, rational homotopy theory provides a natural framework to study these groups, where topological spaces are replaced by commutative differential graded algebras and topological fibrations replaced by algebraic fibrations.

For a CW-complex X, an element α∈πnX is a Gottlieb element if there is a

continuous map H:Sn×X→X such that the following diagram commutes:

where h:Sn→X is a representative map of α and ▽ is the folding map. Moreover, the set of all Gottlieb elements in πnX is a subgroup of πnX denoted by GnX and is called the n-th Gottlieb group of X or the n-th evaluation subgroup of πnX. Alternately, GnX is the image of the map induced on homotopy groups by the evaluation map, ev:aut1X→X, where aut1X denotes the monoid of self-homotopy equivalences of X [3].

Similarly, if p:X→Y is a based map of simply connected CW-complexes and map(X,Y;p), the space of maps from X to Y which are homotopic to p, then the n-th evaluation subgroup of p, also called the n-th generalized evaluation subgroup, is defined in [9] by:

The n-th evaluation subgroup GnX occurs as the special case in which X=Y and p= Id_X. The generalized evaluation subgroups play a well-known role in fixed point theory. In order to apply the generalized evaluation subgroups to fixed point theory, we need the computation of GnY,X;p which are proper subgroups of πnY and contain GnY properly. Unfortunately, there are not many explicit computations of GnY,X;p in literature.

In [10], K.Y. Lee and M.H. Woo introduced the n-th relative evaluation subgroup GnrelY,X;p, also called the n-th relative Gottlieb group, and showed that they fit in a sequence, called G-sequence, which is not necessarily exact:

The exactness of the G-sequence plays an important role in computing homotopy groups.

The complex Steifel and Grassmann form a very well-studied and interesting class of manifolds. They appear abundantly in geometry and topology. Here, we recall that, for 1≤k<n,

where Un is the unitary group [[2], Example 1.83 and Example 1.84]. There is a fibration Uk→iVk,k+nℂ→pGrk,k+nℂ for 1≤k<n.

In this paper, we use the notion of Sullivan minimal model and derivation to determine the rational evaluation subgroups and the rational relative evaluation subgroups of the fibration p:Vk,k+nℂ→Grk,k+nℂ.

This section cannot provide and is not intended to give an introduction to the theory. We expect the reader to have gained a certain familiarity with necessary concepts for example from [1] or [2]. We merely recall some tools and aspects which play a larger role in the paper. All our spaces will be simply connected with the homotopy type of CW-complex with rational cohomology of finite type.

Definition 2.1. A commutative differential graded algebra (cdga) is a graded algebra A=⊕i≥0Ai with a differential d:Ai→Ai+1 such that d2=0, xy=−1ijyx, and dxy=dxy+−1ixdy for all x∈Ai and y∈Aj. A morphism f:(A,d)→(B,d) of cdga's is called a quasi-isomorphism if H∗(f) is an isomorphism. a cdga (A,d) is called simply connected if H0(A)=ℚ and H1(A)=0.

A commutative graded algebra A is free if it is of the form

where Veven=⊕i≥1V2i and Vodd=⊕i≥0V2i+1. A Sullivan algebra is a cgda (ΛV,d), where V=⊕i≥1Vi admits a homogeneous basis {xi}i∈I indexed by a well ordered set I such dxi∈Λ({xi})i<j. A Sullivan algebra is called minimal if dV⊂Λ≥2V. If there is a quasi-isomorphism f:(ΛV,d)→(A,d), where (ΛV,d) is a minimal Sullivan algebra, then we say that (ΛV,d) is a minimal Sullivan algebra of (A,d).

To a simply connected topological space X of finite type, Sullivan associates in a functorial way a cdga APL(X) of piecewise linear forms on X such that H∗(APL(X))≅H∗(X;ℚ) [8]. A Sullivan minimal model of X is a Sullivan minimal model of APL(X). Moreover, the rational homotopy type of X is completely determined by its Sullivan minimal model ΛV,d. In particular, there are isomorphisms

A fibration p:X→Y of simply connected CW-complexes with fiber F has a Sullivan model which is an inclusion: ΛW,d→ΛW⊗ΛV,D of cdga in which ΛW,d is a Sullivan minimal model of Y and ΛW⊗ΛV,D is a Sullivan model (not necessarily minimal) of the total space X.

Definition 2.2. Let ϕ:ΛW,dW→ΛV,dV be a morphism of cdga's. Define a ϕ-derivation 𝜃 of degree n to be a linear map θ:ΛW→ΛV that reduces degree by n such that θxy=θxϕy+−1nxϕxθy. When n=1 we require additionally that dV∘θ=−θ∘dW. Let DernΛW,ΛV;ϕ denote the vector space of ϕ-derivations of degree n for n>0. Define a linear map ∂:DernΛW,ΛV;ϕ→Dern−1ΛW,ΛV;ϕ by ∂θ=dV∘θ−−1θθ∘dW.

Note that Der∗ΛW,ΛV;ϕ,∂ is a chain complex, where Der∗ΛW,ΛV;ϕ=⊕nDernΛW,ΛV;ϕ. In case ΛW≅ΛV and ϕ=IdΛV, the chain complex of derivations Der∗ΛV,ΛV;IdΛV 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

The detailed discussion of the following are in [4]. The post-composition with the augmentation ε:ΛV→ℚ gives a chain complex map

The n-th evaluation subgroup of ϕ is defined as follows:

Then w∗∈HomnW,ℚ(w∗ is the dual of the basis element w of Wn) is in GnΛW,ΛV;ϕ if and only if w∗ extends to a derivation 𝜃 of DernΛW,ΛV;ϕ such that ∂θ=0.

In case ΛW≅ΛV and ϕ=IdΛV, we get the Gottlieb group of ΛV,dV, defined as follows:

In particular, if X is a finite CW-complex, then from [4] and [7] we have

We now recall the definition of the mapping cone of a chain map ϕ:A→B.

Definition 2.3. Let ϕ:A,dA→B,dB be a map of differential graded vector spaces. The mapping cone of ϕ denoted by Rel∗ϕ is defined as follows: Relnϕ=An−1⊕Bn with the differential δa,b=−dAa,ϕa+dBb.

Further, define inclusion and projection maps J:Bn→Relnϕ by Jb=0,b and P:Relnϕ→An−1 by Pa,b=a. These yield a short exact sequence of chain complexes

This definition can be applied to the Sullivan model ϕ:ΛW,dW→ΛV,dV of the fibration p:X→Y.

Note that the pre-composition with ϕ give maps

where ε is the augmentation of either ΛV or ΛW. Following G. Lupton and S.B. Smith [4] (see also [11]), we consider the commutative diagram

On passing to homology and using the naturality of the mapping cone construction, we obtain the following homology ladder for n≥2.

The following definition is very useful to compute the relative evaluation subgroups of a map.

Definition 2.4. Suppose ϕ:ΛW,dW→ΛV,dV is a map of cdga's. We define the n-th relative evaluation subgroup of ϕ by:

We end this section by an overriding hypothesis. In general, we assume that all spaces appearing in the sequel are rational simply connected CW-complex and are of finite type.

Let Grk,nℂ be the complex Grassmann manifold and Vk,nℂ the complex Steifel manifold for 1≤k<n. There is a fibration Uki→Vk,k+nℂp→Grk,k+nℂ. Hence for 1≤k<n, a Sullivan minimal model of p is given by

where ϕx2i=0 for i∈1,...,k and ϕy2j+1=z2j+1 for j∈n,...,n+k−1. In this section, we use Sullivan minimal models to compute the evaluation subgroups and the relative evaluation subgroups of the fibration p:Vk,k+nℂ→Grk,k+nℂ.

We begin by the following which we will use in the sequel.

Theorem 3.1. G∗Vk,nℂ=π∗Vk,nℂ for 1≤k<n.

Proof. First, we recall that an H-space X is a space with a multiplication μ:X×X→X that is associative up to homotopy and admits a unit up to homotopy. Secondly, Since the complex Steifel manifold has the rational homotopy type of an H-space, then the multiplication µ provides a composition Sn×X→X×X→X giving G∗Vk,nℂ=π∗Vk,nℂ, as nedeed.

We note that Theorem 3.1 can also be proved from the Sullivan minimal model for Vk,nℂ. We turn now to the evaluation subgroups of the fibration p.

Theorem 3.2. G∗Grk,k+nℂ,Vk,k+nℂ;p=π∗Grk,k+nℂ for 1≤k<n.

Proof. Write the Sullivan minimal model for Vk,k+nℂ [[2], Example 2.40] as

and the Sullivan minimal model for Grk,k+nℂ [[6], Lemma 1] as

where dWx2i=0 for i∈1,...,k and dy2j+1∈Λ≥2x2,...,x2k for j∈n,...,n+k−1. Here, in both Sullivan minimal models subscripts denoting degrees. Let us calculate G∗Grk,k+nℂ,Vk,k+nℂ;p as follows.

Let x2i,1 denote the derivation αi in Der2iΛW,ΛV;ϕ for i∈1,...,k such that αix2i=1 and zero on other generators. Further, let βj=y2j+1,1 in Der2j+1ΛW,ΛV;ϕ for j∈n,...,n+k−1. Since dy2j+1∈Λ≥2x2,...,x2k, a direct computation shows that βj is a ∂-cycle in Der2j+1ΛW,ΛV;ϕ. Furthermore, a simple analysis on the differential prove that βj cannot bound. This means that, for j∈n,...,n+k−1

Next, consider the derivation αi for i∈1,...,k. We see that

So, by the minimality of ΛW,dW and ϕx2i=0, we get αi(dWe)=0 and further ∂(αi)=0. Otherwise, it is easy to check that αi cannot bound for i∈1,...,k. We omit this detail for clarity. Suppose that there is an odd derivation αi′ in DerΛW,ΛV;ϕ such that ∂(αi′)=αi. This means that in particular (∂(αi′))(x2i)=αi(x2i) and hence 0=1. So, this is a contradiction. Hence, we have proved that αi are non zero homology class in H2iDerΛW,ΛV;ϕ for i∈1,...,k. Furthermore, since

We deduce that

We now continue with the main result. We prove the following:

Theorem 3.3. Consider the fibration Vk,k+nℂp→Grk,k+nℂ for 1≤k<n and φ:(ΛW,dW)→(ΛV,dV) its Sullivan minimal model. Then

Proof. First of all, recall that

where ϕx2i=0 for i∈1,...,k and ϕy2j+1=z2j+1 for j∈n,...,n+k−1. As in the Proof of Theorem 3.3, we denote by αi=x2i,1 in Der2iΛW,ΛV;ϕ for i∈1,...,k,βj=y2j+1,1 in Der2j+1ΛW,ΛV;ϕ and θj=z2j+1,1 in Der2j+1ΛV for j∈n,...,n+k−1. Thus, the map ϕ∗:Der∗ΛV→Der∗ΛW,ΛV;ϕ is given on generators by

Further, in Rel∗ϕ∗, one gets

Therefore, by contradiction it is easy to show that 0,αi are non-bounding δ-cycles. Hence, we have for i∈1,...,k

On other hand, we see that

Moreover for degree reason, it is spanned by

However, in Rel∗ϕ∗^, one gets

Therefore, 0,x2i∗ are cycles, which are not boundaries. Combining all the above, we obtain for i∈1,...,k

It follows that

By Theorem 3.1, Theorem 3.2, Theorem 3.3 and the sequence (1.1), we have the exactness of the G-sequence for j∈n,...,n+k−1

and for i∈1,...,k

Remark 3.4. Since V1,nℂ=S2n−1 and Gr1,nℂ=ℂP2n−1 [2]. Then, our results motivated us to extend (up to changing degree) the O. Maphane results [9].

To will illustrate Theorem 3.3, we propose the following example.

Example 3.5. Consider the fibration V2,5ℂp→Gr2,5ℂ. The Sullivan minimal model of Gr2,5ℂ is given by Λx2,x4,y7,y9,d where dx2=dx4=0,dy7=x42−3x22x4+x24 and dy9=4x23x4−3x2x42−x25. Hence a Sullivan minimal model of p, which we denote by

is given on generators by ϕx2=0=ϕx4,ϕy7=z7 and ϕy9=z9.

Let α2=x2,1,α4=x4,1,β7=y7,1,β9=y9,1 in DerΛW,ΛV;ϕ, and θ7=z7,1 and θ9=z9,1 in DerΛV. Hence, we have

Then, a short computation shows that δ0,α2=0=δ0,α4,δθ7,0=0,β7 and δθ9,0=0,β9. It follows that 0,α2 and 0,α4 are non zero homology classes in H∗Relϕ∗. Moreover, since

we conclude that

The author is very grateful to the referee for carefully reading the manuscript and his helpful suggestions.