Rational Cohomology Algebra of Mapping Spaces between Complex Grassmannians
We consider the complex Grassmannian of k-dimensional subspaces of . There is a natural inclusion . Here, we use Sullivan models to compute the rational cohomology algebra of the component of the inclusion in the space of mappings from to for and in particular to show that the cohomology of contains a truncated algebra , where , for and .
The complex Grassmann is the set of k-planes through the origin in . Moreover,where is the unitary group (, chap. 18). There is a canonical inclusion which is induced by defined by .
The study of the rational homotopy type of function spaces started with Thom in the case where the codomain is an Eilennerg–Maclane space . The first description of a Sullivan model of function spaces is due to Haefliger . Moreover, a model of function spaces between complex projective spaces was given by Moller and Raussen using Postkinov tower . However, there is no explicit and complete description of the homotopy type of the component of the inclusion in the space of mappings from to , and . We begin our work with a model of the inclusion.
In our previous work, we studied the rational homotopy of where is the canonical inclusion. We showed among other things that has the rational homotopy type of a product of odd spheres. Under some assumptions on r, the cohomology algebra of contains either a polynomial algebra or a truncated algebra over a generator of degree 2.
In this paper, we consider the inclusion , .
2. Sullivan Models
Henceforth, we work on the field of rational numbers, .
Definition 1. A differential graded algebra is a graded algebra endowed with a derivation d, of degree , such that . The pair is called a cochain algebra. A graded algebra A is commutative if for (, chap. 3).
Definition 2. A Sullivan algebra is a commutative cochain algebra of the form where and , such that . A Sullivan model for a commutative cochain algebra is a quasi-isomoprphism from a Sullivan algebra . A Sullivan algebra (or model) is said to be minimal if the differential is decomposable, that is, .
Moreover, if , then has a minimal model which is unique up to isomorphism. If X is a nilpotent space and the commutative differential graded algebra () of piecewise linear forms on X, then a Sullivan model of X is a Sullivan model of (, chap.12). A space X is formal if there is a quasi-isomorphism . Moreover, complex Grassmann manifolds are formal .
The cohomology ring of has a presentationa quotient of the polynomial ring generated by , , modulo the ideal generated by the elements . Here, is defined as the degree term in Taylor’s expansion of , where is the total Chern class and for .
For instance, , where and . As forms a regular sequence, the Sullivan model of is hence given by with , and .
3. L∞ Models of Function Spaces
Definition 3. On a graded vector space L, an structure, usually denoted by , is a collection of linear maps, called brackets, where such that the following conditions are satisfied:(1) are graded skew symmetric, that is, for any k permutation σ, where is the sign given by the Koszul convention.(2)The generalised Jacobi identity holds, that is,where denotes the shuffles which are permutations such that and .
An algebra is minimal if . structures are in one-to-one correspondence with codifferentials on the non unital, free commutative coalgebra in which s denotes , that is .
Note that if L is a graded vector space of finite type, an structure on L induces a commutative differential graded algebra structure , , where is defined by and . This is a generalisation of the Quillen cochain functor to algebras (, Chap.23).
Further, an algebra L is an model of a simply connected space X, if is a Sullivan model of X .
Definition 4. Let be a morphism of cochain algebras. A ϕ-derivation θ of degree n is a linear map such that . We denote by the vector space of all ϕ-derivations of degree n. Define a differential of chain complexes , by .
If is a morphism of commutative differential graded algebras, then there is an isomorphism of vector spaces:Moreover, if is a basis for V, we will denote by the ϕ-derivation θ such thatWe now follow  for the definition below.
Consider positive derivations defined byIf is a Sullivan algebra, then given and , of degrees , we define their brackets of length j, bywhere and ϵ is the sign given by the Koszul convention. These operations may be to define a set of linear maps each of degree on as follows:
For ,For , defineby and .
Let be a continuous function between 1-connected spaces and the component of f in the space of continuous maps from X to Y. Let be a model of f. of  proves that is an model of
4. Inclusion of in
Theorem 1. Let be the canonical inclusion. If , then the rational cohomology algebra of contains a truncated algebra , where .
Proof. The cohomology algebras of and are given by and , respectively. Recall that is a smooth manifold of dimension ; hence, for .
The minimal model of iswith , , , and . A model of the inclusion is given bywhere , , , and . Let . Consider . Then , , and are polynomials of degree at least . If , then , , and will be of degree . Therefore, for . In , let . Then for .
Theorem 2. Consider the inclusion . If , then the cohomology algebra of contains a polynomial algebra over a generator of degree 2.
Proof. Letbe a model of the inclusion. Assume , in particular, if , then any odd derivation is of degree at least . Hence, . As , then the derivation is a cycle.
It is sufficient to show that for .
We consider the case when is even, the other case is dealt with in a similar way. For , . Assume , and letThe polynomials , and t are of total degree at least . Assume now that , then . Therefore, for all . Hence, contains a polynomial algebra over a generator of degree 2.
5. The General Case
We can generalise the above results.
Theorem 3. If , then the rational cohomology algebra of contains a truncated algebra where , for and .
Proof. Consider . Letbe the minimal Sullivan model of with , . Moreover,A Sullivan model of is given by , where , and .
Note that the lowest odd degree derivation is of degree as ; then, . Define . Then is a cycle. Moreover,are polynomials of degree . As is of dimension , if , then are of degree .
Therefore, for . In , let . Then for .
Theorem 4. If , then the cohomology algebra of contains a polynomial algebra over a generator of degree 2.
Proof. Letbe a model of the inclusion. By hypothesis, and . Then any odd derivation is of degree at least . Hence, where . As defined in the proof of Theorem 3, consider . It suffices to show that for . Consider even, the other case is dealt with in a similar way.
For , . Assume ,are polynomials of total degree . If , then . Hence, .
This implies that . Therefore, for all . This shows that contains a polynomial algebra over a generator of degree 2.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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