Relative Gottlieb Groups of Embeddings between Complex Grassmannians

Wework in the category of spaces having the homotopy type of simply connected CW complexes of finite type.We denote by h: X⟶ XQ the rationalization of X [1, 2]. Let f: (X, x0)⟶ (Y, y0) be a pointed continuous mapping and map(X, Y; f) be the component of f in the space of all continuous maps g: X⟶ Y. Consider the evaluation map ev: map(X, Y; f)⟶ Y at the base point x0, that is, ev(g) � g(x0). +e nth evaluation subgroup of f, Gn(Y, X; f), is the image of πn(ev) in πn(Y) [3]. In the special case where X � Y and f � 1X, one obtains the Gottlieb group Gn(X) of X [4]. Gottlieb groups play an important role in topology. For instance, if Gn(X) � 0, then any fibration X⟶ E⟶ Sn+1 admits a section (Corollary 2–7 in [4]). In [2], Lee andWoo introduce relative evaluation groups Grel n (Y, X; f) and obtain a long sequence, · · ·⟶ G n+1(Y, X; f)⟶ Gn(X)⟶ Gn(Y, X; f) ⟶ G n (Y, X; f)⟶ · · · , (1) called G-sequence [5]. +is sequence is exact in some cases, for instance, if f is a homotopy monomorphism [6].


Introduction
We work in the category of spaces having the homotopy type of simply connected CW complexes of finite type. We denote by h: X ⟶ X Q the rationalization of X [1,2]. Let f: (X, x 0 ) ⟶ (Y, y 0 ) be a pointed continuous mapping and map(X, Y; f) be the component of f in the space of all continuous maps g: X ⟶ Y. Consider the evaluation map ev: map(X, Y; f) ⟶ Y at the base point x 0 , that is, ev(g) � g(x 0 ). e nth evaluation subgroup of f, G n (Y, X; f), is the image of π n (ev) in π n (Y) [3]. In the special case where X � Y and f � 1 X , one obtains the Gottlieb group G n (X) of X [4]. Gottlieb groups play an important role in topology. For instance, if G n (X) � 0, then any fibration X ⟶ E ⟶ S n+1 admits a section (Corollary 2-7 in [4]).
In [2], Lee and Woo introduce relative evaluation groups G rel n (Y, X; f) and obtain a long sequence, called G-sequence [5]. is sequence is exact in some cases, for instance, if f is a homotopy monomorphism [6].

Rational Relative Gottlieb Groups
e rationalization h: Y ⟶ Y Q induces a rationalization h * : map(X, Y; f) ⟶ map(X, Y; h ∘ f) [7]. erefore, ev * π * (map(X, Y; f)) ⊗Q � ev * π * map X, Y Q ; h∘ f . (2) In this paper, we study the G-sequence of the natural inclusion Gr(k, n) ⟶ Gr(k, n + r) using models of function spaces in rational homotopy [8,9]. In particular, we show that the G-sequence is exact if r ≥ k(n − k). We work with algebraic models in rational homotopy theory introduced by Sullivan and Quillen [10,11]. In this section, we give relevant definitions and fix notation. Details can be found in [1]. All vector spaces and algebras are over the field of rational numbers Q.
Let (A, d) be a cochain algebra. e degree of an homogeneous element a ∈ A p is written |a|. We assume that Moreover, a Sullivan algebra Definition 2. If X is a simply connected space of finite type, then the (minimal) Sullivan model of X is the (minimal) Sullivan model of cdga A PL (X) of polynomial differential forms on X [1,5]. A simply connected topological space X is called formal if there exists a quasi-isomorphism  [11,12]). As G(k, n) � G(n − k, n), we will assume that k ≤ n/2. As G(k, n) is a formal, its Sullivan model can be computed from its cohomology algebra. Precisely, where h j is the polynomial of degree 2j in the Taylor expansion of the expression 1/(1 + x 2 + · · · + x 2k ) [13]. A Sullivan model is given by where dx 2i � 0 and dx 2n− 2k+2i− 1 � h n− k+i , i � 1, . . . , k. Moreover, this model is minimal. Let be respective minimal Sullivan models of Gr(k, n + r) and Gr(k, n). A Sullivan model of the inclusion i: Gr(k, n) ⟶ Gr(k, n + r) is then which is defined by where p ij is a polynomial of degree 2(r + i − j) in y 2 , . . . , y 2k , for i, j � 0, 1, 2, . . . , k − 1, provided that r + i − j ≥ 0. e polynomials p ij encode the relationships between h i 's. ey can be explicitly expressed from the equality: For instance, for k � 2, Example 1. e inclusion Gr(2, 4) ⟶ Gr(2, 7) has a Sullivan model: where We note that − y 2 2 y 4 + y 2 4 � d(y 2 y 5 + y 7 ); therefore, ϕ x 13 � d y 2 y 5 + y 7 y 5 + d y 5 y 7 .
Recall that if ϕ: is a map of chain complexes; the mapping cone of ϕ, denoted by Rel(ϕ), is defined by where the differential is defined by [9] or p. 46 in [14]. Define chain maps J: B n ⟶ Rel n (ϕ) and P: Rel n (ϕ) ⟶ A n− 1 by J(b) � (0, b) and P(sa, b) � a. ere is an exact sequence of chain complexes: 2 International Journal of Mathematics and Mathematical Sciences J Rel * (ϕ) ⟶ P A * − 1 ⟶ 0, (14) which induces a long exact sequence: (see Proposition 4.3 in [14]).
We will restrict to derivations of positive degree; however, in degree one, we only consider those derivations which are cycles.
and zero on other elements of the basis.
Define the Gottlieb group of (∧V, d): en, rational evaluation subgroups are corresponding images in the lower ladder induced in homology by vertical maps. erefore, there is a long sequence: We will use the following result for our computations ( eorem 2.1 in [9] or Corollary 1 in [15]).
We consider the particular case, where f is the inclusion i: Gr(k, n) ⟶ Gr(k, n + r), where r ≥ 1 and its Sullivan model ϕ: (∧V, d) ⟶ (∧W, d) as given in equation (6).

Inclusion Gr(k, n) ⟶ Gr(k, n + 1)
In the range 1 ≤ r < k(n − k), the G-sequence of the inclusion Gr(k, n) ⟶ Gr(k, n + r) is more challenging to characterize, as shown in the following example.