how to understand the galois group of q-bar over q? we want to analyze its action on the nonsingular complex varieties defined over finite extensions of q. this action preserves the underlying etale homotopy type but permutes the manifold structures over it. in 1970, sullivan proposed that there is an abelianized galois symmetry on higher dimensional simply-connected top manifolds by the adams conjecture and it is compatible with the galois symmetries on varieties. it is still an ongoing project to describe this mysterious galois symmetry in a more geometric way by branched coverings. indeed, this agrees with grothendieck's discussion of dessin d'enfants on riemann surfaces in the 1980's. i will report our ongoing works on a generalization to higher dimensions.