Article
Kyungpook Mathematical Journal 2023; 63(1): 131-139
Published online March 31, 2023
Copyright © Kyungpook Mathematical Journal.
Evaluation Subgroups of Mapping Spaces over Grassmann Manifolds
Abdelhadi Zaim
Department of Mathematics and Computer Sciences, Faculty of Sciences Ain Chock, University Hassan II, Casablanca, Morocco
e-mail : abdelhadi.zaim@gmail.com
Received: November 13, 2021; Revised: May 9, 2022; Accepted: May 9, 2022
Abstract
Let
Keywords: Rational homotopy theory, Sullivan minimal model, Gottlieb groups, evaluation subgroups, G-sequence, derivation
1. Introduction
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
continuous map
where
Similarly, if
The n-th evaluation subgroup
In [10], K.Y. Lee and M.H. Woo introduced the n-th relative evaluation subgroup
The exactness of the
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
where
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
2. Preliminaries in Rational Homotopy Theory
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 commutative graded algebra
where
To a simply connected topological space
A fibration
Definition 2.2. Let
Note that
The detailed discussion of the following are in [4]. The post-composition with the augmentation
The n-th evaluation subgroup of ϕ is defined as follows:
Then
In case
In particular, if
We now recall the definition of the mapping cone of a chain map
Definition 2.3. Let
Further, define inclusion and projection maps
This definition can be applied to the Sullivan model
Note that the pre-composition with ϕ give maps
where
On passing to homology and using the naturality of the mapping cone construction, we obtain the following homology ladder for
The following definition is very useful to compute the relative evaluation subgroups of a map.
Definition 2.4. Suppose
We end this section by an overriding hypothesis. In general, we assume that all spaces appearing in the sequel are
3. Evaluation Subgroups of Mapping Spaces over Grassmann Manifolds
Let
where
We begin by the following which we will use in the sequel.
Theorem 3.1.
We note that Theorem 3.1 can also be proved from the Sullivan minimal model for
Theorem 3.2.
and the Sullivan minimal model for
where
Let
Next, consider the derivation
So, by the minimality of
We deduce that
We now continue with the main result. We prove the following:
Theorem 3.3. Consider the fibration
where
Further, in Rel
Therefore, by contradiction it is easy to show that
On other hand, we see that
Moreover for degree reason, it is spanned by
However, in Rel
Therefore,
It follows that
By Theorem 3.1, Theorem 3.2, Theorem 3.3 and the sequence (1.1), we have the exactness of the
and for
Remark 3.4. Since
To will illustrate Theorem 3.3, we propose the following example.
Example 3.5. Consider the fibration
is given on generators by
Let
Then, a short computation shows that
we conclude that
Acknowledgements.
The author is very grateful to the referee for carefully reading the manuscript and his helpful suggestions.
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