  Let X be a smooth complex Fano 4-fold. We show that if X has a small
elementary contraction, then the Picard number rho(X) of X is at most 12. This
result is based on a careful study of the geometry of X, on which we give a lot
of information. We also show that in the boundary case rho(X)=12 an open subset
of X has a smooth fibration with fiber the projective line. Together with
previous results, this implies if X is a Fano 4-fold with rho(X)>12, then every
elementary contraction of X is divisorial and sends a divisor to a surface. The
proof is based on birational geometry and the study of families of rational
curves. More precisely the main tools are: the study of families of lines in
Fano 4-folds and the construction of divisors covered by lines, a detailed
study of fixed prime divisors, the properties of the faces of the effective
cone, and a detailed study of rational contractions of fiber type.
