arXiv subjects
Publications and source records attributed to Michael Falk.
In "On the homotopy theory of arrangements," published in 1986, the authors gave a comprehensive survey of the subject. This article updates and continues the earlier article, noting some key open problems.
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 $\C^r\setminus \bigcup \A.$ 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 $M'_n$, $n\geq 1$, 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 $§$ denote the arrangement consisting of the hyperplane $\{0\}$ in $\C^1$. 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 $\A_0 \oplus \A_1$ and $§\oplus P(\A_0,\A_1)$ have diffeomorphic complements.
We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes $\A$. The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only non-vanishing cohomology group, under some nonresonance conditions on the local system, for any arrangement $\A$. The bases are determined by a linear ordering of the hyperplanes, and are indexed by certain ``no-broken-circuits" bases of $\A$. The basic forms depend on the local system, but any two bases constructed in this way are related by a matrix of integer constants which depend only on the linear orders and not on the local system. In certain special cases we show the existence of bases of monomial logarithmic forms.