Rational formality of function spaces (Q2381425)

Consider the mapping space map\((X,Y)\) in which both \(X\) and \(Y\) are CW-complexes of finite type, \(X\) is nilpotent and the rational dimension of \(X\) is smaller than the connectivity of \(Y\) (so that map(X,Y) is path connected). Assume also that the rational Hurewicz morphism of \(X\) is non zero in for some odd dimensional rational homotopy group, and that the rational cohomology algebra of \(Y\) is finitely generated. Then, the following theorem is proved: If map\((X,Y)\) is formal, (i.e. its rational homotopy type depends only on its rational cohomology algebra) then \(Y\) has the rational homotopy type of a finite product of Eilenberg Mac Lane spaces. To show that the first assumption is necessary the author also proves that map\(\bigl(S^2,K({\mathbb Z},4)\vee K({\mathbb Z},4)\bigr)\) is a formal space.