Relative Gottlieb Groups of Embeddings between Complex Grassmannians

Abstract

Let be the complex Grassmann manifold of -linear subspaces in . We compute rational relative Gottlieb groups of the embedding and show that the -sequence is exact if .

1. Introduction

We work in the category of spaces having the homotopy type of simply connected CW complexes of finite type. We denote by the rationalization of [1, 2]. Let be a pointed continuous mapping and be the component of in the space of all continuous maps . Consider the evaluation map at the base point , that is, . The th evaluation subgroup of , , is the image of in [3]. In the special case where and , one obtains the Gottlieb group of [4]. Gottlieb groups play an important role in topology. For instance, if , then any fibration admits a section (Corollary 2–7 in [4]).

In [2], Lee and Woo introduce relative evaluation groups and obtain a long sequence,called G-sequence [5]. This sequence is exact in some cases, for instance, if is a homotopy monomorphism [6].

2. Rational Relative Gottlieb Groups

The rationalization induces a rationalization [7]. Therefore,

In this paper, we study the -sequence of the natural inclusion using models of function spaces in rational homotopy [8, 9]. In particular, we show that the -sequence is exact if . 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 .

Let be a cochain algebra. The degree of an homogeneous element is written . We assume that is 1-connected, that is, and . The algebra is called commutative if for homogeneous elements .

Definition 1. A commutative differential graded algebra (cdga, for short) is called a Sullivan algebra if , where . It will be denoted by .
Moreover, a Sullivan algebra is called minimal if . A Sullivan model of is given by a Sullivan algebra together with a quasi-isomorphism . It is unique up to isomorphism.

Definition 2. If is a simply connected space of finite type, then the (minimal) Sullivan model of is the (minimal) Sullivan model of cdga of polynomial differential forms on [1, 5]. A simply connected topological space is called formal if there exists a quasi-isomorphism , where is a Sullivan model of . Formal spaces include homogeneous spaces , where and have the same rank.

The complex Grassmann manifold is the space of -dimensional subspaces of . Moreover, , where is the unitary group. Hence, is formal (see also [11, 12]). As , we will assume that . As is a formal, its Sullivan model can be computed from its cohomology algebra. Precisely,where is the polynomial of degree in the Taylor expansion of the expression [13]. A Sullivan model is given bywhere and , . Moreover, this model is minimal.

Letbe respective minimal Sullivan models of and . A Sullivan model of the inclusion is thenwhich is defined bywhere is a polynomial of degree in , for , provided that .

The polynomials encode the relationships between ’s. They can be explicitly expressed from the equality:

For instance, for ,

Example 1. The inclusion has a Sullivan model:whereWe note that ; therefore,

Recall that if is a map of chain complexes; the mapping cone of , denoted by , is defined bywhere the differential is defined by [9] or p. 46 in [14]. Define chain maps and by and . There is an exact sequence of chain complexes:which induces a long exact sequence:(see Proposition 4.3 in [14]).

Definition 3. Let be a morphism of cdga’s. A -derivation of degree is a linear mapping such that . We denote by the vector space of -derivations of degree and by the -graded vector space of all -derivations. The differential on is defined by . We will restrict to derivations of positive degree; however, in degree one, we only consider those derivations which are cycles.

If is the identity mapping, we simply write for . Moreover, if , where is a basis of and is a morphism of cdga’s, we denote by the unique -derivation such that and zero on other elements of the basis.

Define the Gottlieb group of :

Hence, , where is the postcomposition with the augmentation map . If is simply connected and is the minimal Sullivan model of , then , where is the rationalization (Propostion 29.8 in[1]).

Similarly, if is a map between Sullivan algebras, then the Gottlieb group is defined as , where is the postcomposition with . Moreover, if is a Sullivan model of a map , where is finite, then , where is the rationalization map.

Let be a Sullivan model of a map between simply connected spaces. It induces a chain map by precomposition by . We get the following commutative diagram:

Then, rational evaluation subgroups are corresponding images in the lower ladder induced in homology by vertical maps. Therefore, there is a long sequence:

We will use the following result for our computations (Theorem 2.1 in [9] or Corollary 1 in [15]).

Theorem 1 (see [9]). Let be a map between simply connected CW complexes, where is of finite type and its Sullivan model. The long exact sequence induced by the map on rational homotopy groups is equivalent to the long exact sequence of

We consider the particular case, where is the inclusion , where and its Sullivan model as given in equation (6).

Theorem 2. Let be a Sullivan model of the inclusion , where :(1), the dual of(2)

Proof. (1)Recall that , , and are defined by , , and is a polynomial of degree in and . We consider the composition . As is a quasi-isomorphism, then the -sequence of the inclusion is computed from the long exact sequence induced by the cone of the map: Each of the derivations is a cycle of degree at least and cannot be boundary as all even degree derivations in are of degree at most . Hence, is nonzero in  Consider the derivations , for . Then, Moreover, as , then . Therefore, Therefore, . Hence, is a cycle for . Moreover, cannot be a boundary as all odd degree derivations are of degree at least . Therefore, are cycles which cannot be boundaries for degree reasons. Hence, .(2)First, we note that , and consequently, [1, 16]. Moreover, a straightforward calculation shows thatWe consider the vector space:where the differential is defined by . Consider in . For degree reasons, . Therefore, , for . Hence, represent nonzero homology classes in . We conclude that .

Corollary 1. If , then the rational -sequence of the inclusion is exact.

Proof. It comes from the previous lemma that the -sequence iswhich is exact.

3. Inclusion

In the range , the -sequence of the inclusion is more challenging to characterize, as shown in the following example.

Example 2. Consider the inclusion of which a Sullivan model is given bywhere is defined in Example 1. We compose with the quasi-isomorphism and consider , where . Moreover, , where and similarly . Furthermore, ; hence, represents a nonzero homology class in . A simple calculation shows that , where . Hence,ConsiderThen,where and . Therefore, the image ofis zero. Hence, the sequencereduces towhich is not exact.
In the same way,is not exact. Moreover, is an isomorphism.

Although the -sequence of the inclusion might not be exact for some values of , we have the following result for .

Theorem 3. Let be a Sullivan model of the inclusion :(1) has dimension 1(2)The -sequence of the inclusion is not exact

Proof. Recall from Section 2 that the minimal Sullivan model of is , whereSimilarly, a model of is , whereMoreover, a model of the inclusion is given by and defined byWe consider the quasi-isomorphismand set . ConsiderWe have the following relations:As a result, inwe have the following relations:We consider the commutative diagram:Let and . ConsiderThen,Hence,Moreover, the image of is . Therefore,This shows the first part of the theorem and corrects Theorem 3 in [17] and Theorem 3 [18].
Moreover,is the zero map. The -sequence then reduces to exact portionsfor , andand a nonexact part,

Example 3. We consider a model of the inclusion which is of the formwhere , , , , , , , , , and .

We compose which the quasi-isomorphism to get . Inwe have the following relations:

Considerwhere

Hence,

However, . Therefore, . As and , then the -sequence reduces to exact nonzero fragments:and a nonexact sequence,

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The author declares that there are no conflicts of interest.