Non-Zero Degree Maps Between 2n-Manifolds

ThomPontrjagin constructions are used to give a computable necessary and sufficient condition for a homomorphism : H n(L;Z) H n(M;Z) to be realized by a map f : M L of degree k for closed (n 1)-connected rn-manifolds M and L, n > 1. A corollary is that each (n 1)-connected 2n-manifold admits selfmaps of degree larger than 1, n > 1. In the most interesting case of dimension 4, with the additional surgery arguments we give a necessary and sufficient condition for the existence of a degree k map from a closed orientable 4-manifold M to a closed simply connected 4-manifold L in terms of their intersection forms; in particular, there is a map f : M L of degree 1 if and only if the intersection form of L is isomorphic to a direct summand of that of M.