  We apply spectral sequences to derive both an obstruction to the existence of
$n$-fold prolongations and a topological classification. Prolongations have
been used in the literature in an attempt to prove that every Engel structure
on $M\times\mathbb{S}^1$ with characteristic line field tangent to the fibers
is determined by the contact structure induced on a cross section and the
twisting of the Engel structure along the fibers. Our results show that this
statement needs some modification: to classify the diffeomorphism type of the
Engel structure we additionally have to fix a class in the first cohomology of
$M$.
