Rational function spaces (Q1429080)

In this paper, the author studies first, the number of the path components of \(\text{Map}(X,Y)\), when \(Y\) is the pullback of a map between two loop spaces (generalizing a result of Spanier) and second, the rational homotopy type of the path-components of \(\text{Map}(X,Y)\). The author determines the number of path-components of \(\text{Map}(X,Y)\) for \(Y\) being the total space of a principal fibration induced by a map between two loop spaces. In particular, he computes the number of the path-components of \(\text{Map}(X,\mathbb{C} P^{n})\) and \(\text{Map} (X,S^{2n})\) rationally. He generalizes the result of Moller and Raussen determining the distinct rational homotopy types of the components of the mapping spaces. As a corollary, the author gives a complete answer to determine when \(\text{Map}_{f} (X_{1},Y)\) is rationally homotopy equivalent to \(\text{Map}_{g} (X_{2},Y)\), for \(Y\) a complex projective space and \(X_{1},X_{2}\) simply connected spaces.