  Let \A be a complex hyperplane arrangement, and let $X$ be a modular element
of arbitrary rank in the intersection lattice of \A. We show that projection
along $X$ restricts to a fiber bundle projection of the complement of \A to the
complement of the localization $\A_X$ of \A at $X$. The fiber is the decone of
a realization of the complete principal truncation of the underlying matroid of
\A along the flat corresponding to $X$. This result gives a topological
realization of results of Stanley, Brylawsky, and Terao on modular
factorization. We show that (generalized) parallel connection of matroids
corresponds to pullback of fiber bundles, clarifying the notion that all
examples of diffeomorphisms of complements of inequivalent arrangements result
from the triviality of the restriction of the Hopf bundle to the complement of
a hyperplane. The modular fibration result also yields a new method for
identifying $K(\pi,1)$ arrangements of rank greater than three. We identify a
new families of $K(\pi,1)$ arrangements, providing more evidence for the
conjecture that factored arrangements of arbitrary rank are $K(\pi,1)$.
