  This paper provides some technical results needed in "Formalism for Relative
Gromov-Witten Invariants." We study line-bundles on the moduli stacks of
relative stable and rubber maps that are used to define relative Gromov-Witten
invariants in the sense of J. Li. Relations between these line-bundles yield
degeneration formulae. In addition we prove a technical result called the
Trivial Cylinders Theorem.
