Problem 4. (Covering map) We say a smooth map T : X - Y is a covering map if a is surjective and for every p E M there exists a neighbourhood U of p such that each connected component of -1(U) is mapped diffeomorphically onto U by T. (1) Show that a covering map 4 : X - Y is a local diffeomorphism. (2) Show that an injective covering map 7 : X - Y is a diffeomorphism. (3) Show that 7 : R2 -> T2 defined in Problem 3 (2) is a covering map. (4) Show that if m : X1 - Y1 and 12 : X2 - Y2 are covering maps, then T1 X 12 : X1 X X2 -> Yi X Y2 is a covering map. (5) Suppose T : X - Y is a proper local diffeomorphism, then T is a covering map. (6) Given an example of a local diffeomorphism that is not a covering map