TY - JOUR

T1 - Parallel connections and bundles of arrangements

AU - Falk, Michael J.

AU - Proudfoot, Nicholas J.

N1 - Funding Information:
✩Research conducted in part under an NSF Research Experiences for Undergraduates grant.

PY - 2002/2/28

Y1 - 2002/2/28

N2 - Let A be a complex hyperplane arrangement, and let X be a modular element of arbitrary rank in the intersection lattice of A. Projection along X restricts to a fiber bundle projection of the complement of A to the complement of the localization AX of A at X. We identify the fiber as the decone of a realization of the complete principal truncation of the underlying matroid of A along the flat corresponding to X. We then generalize to this setting several properties of strictly linear fibrations, the case in which X has corank one, including the triviality of the monodromy action on the cohomology of the fiber. This gives a topological realization of results of Stanley, Brylawsky, and Terao on modular factorization. We also 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 theorem also yields a new method for identifying K(π,1) arrangements of rank greater than three. We exhibit new families of K(π,1) arrangements, providing more evidence for the conjecture that factored arrangements of arbitrary rank are K(π,1).

AB - Let A be a complex hyperplane arrangement, and let X be a modular element of arbitrary rank in the intersection lattice of A. Projection along X restricts to a fiber bundle projection of the complement of A to the complement of the localization AX of A at X. We identify the fiber as the decone of a realization of the complete principal truncation of the underlying matroid of A along the flat corresponding to X. We then generalize to this setting several properties of strictly linear fibrations, the case in which X has corank one, including the triviality of the monodromy action on the cohomology of the fiber. This gives a topological realization of results of Stanley, Brylawsky, and Terao on modular factorization. We also 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 theorem also yields a new method for identifying K(π,1) arrangements of rank greater than three. We exhibit new families of K(π,1) arrangements, providing more evidence for the conjecture that factored arrangements of arbitrary rank are K(π,1).

KW - Fibration

KW - K(π,1) arrangement

KW - Matroid

KW - Modular element

KW - Parallel connection

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U2 - 10.1016/S0166-8641(01)00042-6

DO - 10.1016/S0166-8641(01)00042-6

M3 - Article

AN - SCOPUS:0038690546

SN - 0166-8641

VL - 118

SP - 65

EP - 83

JO - Topology and its Applications

JF - Topology and its Applications

IS - 1-2

ER -