  Let X be a smooth complex Fano variety. We define and study 'quasi
elementary' contractions of fiber type f: X -> Y. These have the property that
rho(X) is at most rho(Y)+rho(F), where rho is the Picard number and F is a
general fiber of f. In particular any elementary extremal contraction of fiber
type is quasi elementary. We show that if Y has dimension at most 3 and Picard
number at least 4, then Y is smooth and Fano; if moreover rho(Y) is at least 6,
then X is a product. This yields sharp bounds on rho(X) when dim(X)=4 and X has
a quasi elementary contraction, and other applications in higher dimensions.
