Orlik-Solomon Algebras and Tutte Polynomials

Abstract

The OS algebra A of a matroid M is a graded algebra related to the Whitney homology of the lattice of flats of M. In case M is the underlying matroid of a hyperplane arrangement A in ℂr , A is isomorphic to the cohomology algebra of the complement ℂr ∖∪A. Few examples are known of pairs of arrangements with non-isomorphic matroids but isomorphic OS algebras. In all known examples, the Tutte polynomials are identical, and the complements are homotopy equivalent but not homeomorphic.We construct, for any given simple matroid M 0, a pair of infinite families of matroids M n and M ′ n , n ≥ 1, each containing M 0 as a submatroid, in which corresponding pairs have isomorphic OS algebras. If the seed matroid M 0 is connected, then M n and M ′ n have different Tutte polynomials. As a consequence of the construction, we obtain, for any m, m different matroids with isomorphic OS algebras. Suppose one is given a pair of central complex hyperplane arrangements A 0 and A 1 . Let S denote the arrangement consisting of the hyperplane {0} in ∪1 . We define the parallel connection P(A 0, A 1), an arrangement realizing the parallel connection of the underlying matroids, and show that the direct sums A 0 ⊕ A 1 and S ⊕ P (A 0, A 1) have diffeomorphic complements.