Rational Cohomology Algebra of Mapping Spaces between Complex Grassmannians

We consider the complex Grassmannian Gr(k, n) of k-dimensional subspaces of C. 'ere is a natural inclusion in,r: Gr(k, n)↪Gr(k, n + r). Here, we use Sullivan models to compute the rational cohomology algebra of the component of the inclusion in,r in the space of mappings from Gr(k, n) to Gr(k, n + r) for r≥ 1 and in particular to show that the cohomology of map(Gr(n, k),Gr(n, k + r); in,r) contains a truncated algebra Q[x]/x 2 − , where |x| � 2, for k≥ 2 and n≥ 4.


Introduction
e complex Grassmann Gr(k, n) is the set of k-planes through the origin in C n . Moreover, where U(n) is the unitary group ( [1], chap. 18). ere is a canonical inclusion i n,r : Gr(k, n) ↪ Gr(k, n + r) which is induced by i r : C n ⟶ C n+r defined by i(x 1 , . . . , x n ) � (x 1 , . . . , x n , 0, . . . , 0). e study of the rational homotopy type of function spaces started with om in the case where the codomain is an Eilennerg-Maclane space [2]. e first description of a Sullivan model of function spaces is due to Haefliger [3]. Moreover, a model of function spaces between complex projective spaces was given by Moller and Raussen using Postkinov tower [4]. However, there is no explicit and complete description of the homotopy type of the component of the inclusion Gr(k, n) ↪ Gr(k, n + r) in the space of mappings from Gr(k, n) to Gr(k, n + r), r ≥ 1 and n ≥ 4. We begin our work with a model of the inclusion.
In our previous work, we studied the rational homotopy of map(Gr(2, n), Gr(2, n + r); i n,1 ) where i n,r : Gr(2, n) ⟶ Gr(2, n + r) is the canonical inclusion. We showed among other things that map(Gr(2, n), Gr(2, n + 1); i n,r ) has the rational homotopy type of a product of odd spheres. Under some assumptions on r, the cohomology algebra of map(Gr(2, n), Gr(2, n + r); i n,r ) contains either a polynomial algebra or a truncated algebra over a generator of degree 2.

Sullivan Models
Henceforth, we work on the field of rational numbers, Q.

Definition 2.
A Sullivan algebra is a commutative cochain algebra of the form Moreover, if H 0 (A) � Q, then (A, d) has a minimal model which is unique up to isomorphism. If X is a nilpotent space and A PL (X) the commutative differential graded algebra (cdga) of piecewise linear forms on X, then a Sullivan model of X is a Sullivan model of A PL (X) ( [6], chap.12). A space X is formal if there is a quasi-isomorphism (∧V, d) ⟶ H * (X, Q). Moreover, complex Grassmann manifolds are formal [7]. e cohomology ring H * (Gr(k, n), Q) of Gr(k, n) has a presentation a quotient of the polynomial ring generated by c 1 , c 2 , . . . , c k , |c i | � 2i, modulo the ideal generated by the elements h j , n − k + 1 ≤ j ≤ n. Here, h j is defined as the 2j th degree term in Taylor's expansion of (1 + c 1 + c 2 + c 3 + · · · + c k ) − 1 , where (1 + c 1 + c 2 + c 3 + · · · + c k ) is the total Chern class and forms a regular sequence, the Sullivan model of Gr(2, 4) is hence given by

L ∞ Models of Function Spaces
Definition 3. On a graded vector space L, an L ∞ structure, usually denoted by (L, l k k≥1 ), is a collection of linear maps, l k k≥1 ) called brackets, where l k : ⊗ k L ⟶ L such that the following conditions are satisfied: (1) l k are graded skew symmetric, that is, for any k permutation σ, where ε σ is the sign given by the Koszul convention. (2) e generalised Jacobi identity holds, that is, where S(i, n − i) denotes the (i, n − i) shuffles which are permutations σ ∈ S n such that σ(1) < · · · < σ(i) and σ(i + 1) < · · · < σ(n).
An L ∞ algebra (L, l k k≥1 ) is minimal if l 1 � 0. L ∞ structures are in one-to-one correspondence with codifferentials on the non unital, free commutative coalgebra Λ + sL in which s denotes suspension, that is (sL) k � L k− 1 [9].
Note that if L is a graded vector space of finite type, an L ∞ structure on L induces a commutative differential graded . is is a generalisation of the Quillen cochain functor to L ∞ algebras ( [6], Chap.23).
Further, an L ∞ algebra L is an L ∞ model of a simply connected space X, if C ∞ (L) is a Sullivan model of X [10].

The General Case
We can generalise the above results.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.