When working with fibrations,
there are times when standard topological constructions involving identifications are
useful. The problem of course, is to show that identifying fibrations in the
proper way yields a fibration. This paper establishes a fairly general result
concerning attaching Hurewicz fibrations over a fixed base space with a fiber
preserving map. This can be applied to obtain many common topological
constructions. In particular a theorem of P. Tulley on mapping cylinders is
strengthened, which in turn strengthens the main theorems on strong fiber homotopy
equivalence and extensions of fibrations obtained by P. Tulley and S. Langston. In
addition, these results are applied to obtain a stronger version of Dold’s
pasting lemma, an important step in the construction of classifying spaces for
fibrations.
