Some Problems In Rational Homotopy Theory

In this thesis we study the rational homotopy theory of the mapping spaces,clas-sifying spaces and intrinsically formal spaces.Our main results are the followings.1.By using homotopy transfer techniques in the context of rational homotopy the-ory,we show that for a finite CW-complex X and a rational finite type CW-complex Y with the minimal Sullivan model of the form(?(P?Q),d P=0,d Q(?)?P),the ratio-nal homotopy type of map(X,Y)is determined by the cohomology algebra H^*(X;Q)and the Sullivan algebra(?(P?Q),d).From this,we show:(1)the existence of H-structures on a component of the mapping space map(X,Y);(2)map(X,Y;f)map(X,Y;f)if f and f are connected by an algebra automorphism of H^*(X,Q).2.We show that the non-trivial rank two rational spaces can not be realized as the classifying space of any rational space with specific properties.We also observe that if Eilenberg-Mac Lane space K(Q²,n)can be realized as the classifying space of a simply connected elliptic rational space X,then X has the rational homotopy type of S^n-1×S^n-1with n even.3.In this thesis,the C_?-algebra is introduced from a rational homotopy point of view.We provide a criteria for establishing the intrinsically formality of a simply connected space with rational cohomology finite type in terms of C_?-algebra.As an application,we show certain types of spaces are intrinsically formal.We prove the followings theorems:(1)every compact simply connected manifold M of dimension less or equal to 6 is intrinsically formal;(2)let X be a n-connected m-formal dimension space with m?3n+1,n?1,then X is intrinsically formal;(3)let M be a(k-1)-connected manifold of dimension less than or equal to(4k-2),then M is intrinsically formal.