Abstract
Let for denote the Grassmann manifold of -dimensional vector subspaces of . In this paper, we compute, in terms of the Sullivan models, the rational evaluation subgroups and, more generally, the -sequence of the inclusion for .
1. Introduction
Throughout this paper, we rely on the theory of minimal Sullivan models in rational homotopy theory for which [1] is our standard reference. Let be a based CW-complex; the th Gottlieb group of (or the th evaluation subgroup [2] of ), denoted by , consists of those elements for which there is a continuous map such that the following diagram commutes:where is a representative of and is the folding map. Let be a based map of simply connected finite CW-complexes. In [3], the evaluation at the base point of gives the evaluation map where is the component of in the space of mappings from to The image of the homomorphism induced in homotopy groupsis called the th evaluation subgroup of p, and it is denoted by Moreover, if the space is the monoid of self-equivalences of homotopic to the identity of then, is the evaluation map, and the image of the induced homomorphismis i.e., the th Gottlieb group. Moreover, in [4], Woo and Lee studied the relative evaluation subgroups and proved that they fit in a sequencecalled the -sequence of . Finally, in [3], Smith and Lupton identify the homomorphism induced on rational homotopy groups by the evaluation map in terms of a map of complexes of derivations constructed directly from the Sullivan minimal model of . In [5], the authors use a map of complexes of derivations of minimal Sullivan models of mapping spaces to compute rational relative Gottlieb groups of the inclusion between complex Grassmannians. In this paper, we generalize their work to compute rational relative Gottlieb groups of the inclusion between complex Grassmannians for .
2. Preliminaries
Here, we fix terminology and recall some standard facts on differential graded algebras. All vector spaces and algebras are taken over a field of rational numbers.
Definition 1. A graded algebra is a sum where is a vector space, together with an associative multiplication , and has It is graded commutative if, for any homogeneous elements and ,where for If is a graded algebra equipped with a linear differential map such that andthen is called a differential-graded algebra and is called a differential. Moreover, if is also a graded commutative algebra, then is a commutative differential graded algebra (cdga). It is said to be connected if .
Definition 2. Let with and A commutative-graded algebra is called free commutative if , where is the symmetric algebra on and is the exterior algebra on .
Definition 3. A Sullivan algebra is a commutative differential-graded algebra , where and such that and It is called minimal if
If is a cdga whose cohomology is connected and finite dimensional in each degree, then there always exists a quasi-isomorphism from a Sullivan algebra to [1]. To each simply connected space, Sullivan associates a cdga of rational polynomial differential forms on that uniquely determines the rational homotopy type of [6]. A minimal Sullivan model of is a minimal Sullivan model of More precisely, as graded algebras and as graded vector spaces.
3. Derivations of a Sullivan Model and the -Sequence
Let be a commutative differential-graded algebra. A derivation of degree is a linear mapping such that Denote by the vector space of all derivation of degree and The commutator bracket induces a graded Lie algebra structure on Moreover, is a differential graded Lie algebra [6], with the differential defined in the usual way by
Let be a Sullivan algebra, where is spanned by Then, is spanned by where is the unique derivation of defined by The derivation will be denoted by Moreover, an element is a Gottlieb element of if and only if there is a derivation of satisfying and such that ,see page 392 in [1].
Let be a morphism of cdga’s. A -derivation of degree is a linear mapping for which
We consider only derivations of positive degree. Denote by the vector space of -derivations of degree for and by the -graded vector space of all -derivations. The differential-graded vector space of -derivations is denoted by where the differential is defined by In case and then is just the usual differential-graded Lie algebra of derivations on the cdga [3]. Whenever is a Sullivan algebra, we note that there is an isomorphism of graded vector spaces:
If is a basis of then the vector space is spanned by the unique -derivation denoted by such that where and for Moreover, in [3], precomposition with gives a chain complex map and postcomposition with the augmentation gives a chain complex map The evaluation subgroup of is defined as follows:
In case and we get the Gottlieb group of defined as follows:
In particular, if is the minimal Sullivan model of a simply connected space (see Proposition 29.8 in [1]).
Definition 4. (see [3, 7]). Let be a map of differential-graded vector spaces. A differential-graded vector space, called the mapping cone of is defined as follows. with the differential There are inclusion and projection chain maps and defined by and These yields a short exact sequence of chain complexesand a long exact homology sequence of whose connecting homomorphism is
Following [3], we consider a commutative diagram of differential-graded vector spaces:where is the augmentation of either or which leads to the following homology ladder for :
The th relative evaluation subgroup of is defined as follows:
The -sequence of the map is given bywhich ends in Moreover, Theorem 3.5 in [3] can be applied to the Sullivan model of the map
4. The Inclusion
The complex Grassmannian is a simply connected homogeneous space as for where is the unitary group. It is a symplectic manifold of dimension where As the complex Grassmannian is simply connected, so we may associate a minimal Sullivan model.
The method to compute a Sullivan model of the homogeneous space is given in detail in [8, 9].
Following [9], a Sullivan model of for is given bywith
Lemma 1. The minimal Sullivan model of for is given bywhere and
Proof. Consider the Sullivan modelof for The model is not minimal as the linear part is not zero. To find its minimal Sullivan model, we make a change of variable and replace by wherever it appears in the differential. This gives an isomorphic Sullivan algebrawhereAs the ideal generated by and is acyclic, the above Sullivan algebra is quasi-isomorphic towhereOne continues in this fashion and makes another change of variable and replaces by wherever it appears in the differential and does so until they reach a change of variable of the formwhere and replacewherever it appears in the differential. This gives an isomorphic Sullivan algebra:whereAs the ideal generated by and is acyclic, we get the minimal Sullivan model:with and
In the same way, by Lemma 1, the minimal Sullivan model of for and is given bywhere and We establish the following results.
Theorem 1. Let Then,
Proof. Let denote the derivation for such that and zero on other generators. Then,Moreover, the generators cannot be boundaries for degree reasons. Therefore, are nonzero homology classes in Furthermore,
As is a simply connected finite CW-complex, then (see Proposition 28.8 in [1]. Thus,
Theorem 2. Given the inclusion for and for its Sullivan model, then
Proof. The vector space is generated by the derivations for The differential is given byHence, are the nonzero homology classes in Moreover, , where Thus,
Theorem 3. Given the inclusion for and for its Sullivan model, then
Proof. Define the derivations for in and in Then,such thatand Thus, if and are the nonzero homology classes in Moreover, , for , and in the same way, A straightforward calculation shows that for and span
The -sequence reduces toand it is exact.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
This work was completed with the support of the Botswana International University of Science and Technology (BIUST).