In 1960, A. Haefliger presented necessary and sufficient conditions for factoring a smooth map from a surface into the plane through an immersion into
R
3
{R^3}
. Here, necessary and sufficient conditions are given for factoring a map between manifolds of dimension n,
n
⩾
2
n \geqslant 2
, through an immersion into a line bundle over the range. In addition, conditions are given for factoring such a map through a submersion from a line bundle over the domain.