  Elementary geometric arguments are used to compute the group of homotopy
classes of maps from a 4-manifold X to the 3-sphere, and to enumerate the
homotopy classes of maps from X to the 2-sphere. The former completes a project
initiated by Steenrod in the 1940's, and the latter provides geometric
arguments for and extensions of recent homotopy theoretic results of Larry
Taylor. These two results complete the computation of all the cohomotopy sets
of closed oriented 4-manifolds and provide a framework for the study of Morse
2-functions on 4-manifolds, a subject that has garnered considerable recent
attention.
