Parallel connections and bundles of arrangements

Michael J. Falk, Nicholas J. Proudfoot

Published 2000-02-12, updated 2000-07-18Version 2

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 $K(\pi,1)$.