Commun. Korean Math. Soc. 37 (2022), No. 1, pp. 259–267 https://doi.org/10.4134/CKMS.c200431 pISSN: 1225-1763 / eISSN: 2234-3024 RATIONAL HOMOTOPY TYPE OF MAPPING SPACES BETWEEN COMPLEX PROJECTIVE SPACES AND THEIR EVALUATION SUBGROUPS Jean-Baptiste Gatsinzi Abstract. We use L∞ models to compute the rational homotopy type of the mapping space of the component of the natural inclusion i n,k : CP n  → CP n+k between complex projective spaces and show that it has the rational homotopy type of a product of odd dimensional spheres and a complex projective space. We also characterize the mapping aut 1 CP n → map(CP n , CP n+k ; i n,k ) and the resulting G-sequence. 1. Introduction Let f : X → Y be a map between simply connected CW-complexes of finite type. We denote by map(X,Y ; f ) the path component of f in the space of continuous mappings from X to Y . The study of the rational homotopy type of map(X,Y ; f ) was initiated by Haefliger [10] who describes its Sullivan model. Afterwards there were attempts to find a Quillen model of map(X,Y ; f ) from either a Sullivan or a Quillen model of f . Chain complexes of which the homology coincides with rational homotopy groups of function spaces were investigated [8,12,13]. Those chain complexes were later developed into models of function spaces [2–5]. Following [5] we describe in this paper an L ∞ model of the inclusion i n,k : CP n  → CP n+k . We shall use rational homotopy theory for which the standard reference is [6]. The notion of L ∞ -algebra was introduced by Lada [11] and we remind here the definition. Definition 1. A permutation σ ∈ S n is called an (i,n − i)-shuffle if σ(1) < ··· <σ(i) and σ(i + 1) < ··· <σ(n), where i =1,...,n. For graded objects x 1 ,...,x n , the Koszul sign ǫ(σ) is determined by x 1 ∧···∧ x n = ǫ(σ)x σ(1) ∧···∧ x σ(n) . It depends not only of the permutation σ but also of degrees of x 1 ,...,x n . Received November 16, 2020; Accepted January 20, 2021. 2010 Mathematics Subject Classification. Primary 55P62; Secondary 54C35. Key words and phrases. Rational homotopy theory, mapping space, L∞ algebra. c 2022 Korean Mathematical Society 259