Orlik-Solomon Algebras and Tutte Polynomials

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 ℂ^r, A is isomorphic to the cohomology algebra of the complement ℂ^r \∪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 M₀, a pair of infinite families of matroids M_n and M′_n, n ≥ 1, each containing M₀ as a submatroid, in which corresponding pairs have isomorphic OS algebras. If the seed matroid M₀ 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 A₀ and A₁. Let S denote the arrangement consisting of the hyperplane {0} in ℂ¹. We define the parallel connection P(A₀, A₁), an arrangement realizing the parallel connection of the underlying matroids, and show that the direct sums A₀ ⊕ A₁ and S ⊕ P (A₀, A₁) have diffeomorphic complements.