The Cohomology of Quaternionic Hyperplane Complements

Abstract

Over the complex numbers, the complement of a collection of hyperplanes is a widely-studied object; the cohomology ring, in particular, is known to have a structure depending only on the combinatorial properties of the intersection of hyperplanes. The fundamental group, on the other hand, requires specific knowledge about the particular embedding in complex space, though the tower of nilpotent quotients can still be determined (see arXiv:math/9805056). Over the quaternions, however, since hyperplane complements are simply connected, the topological properties of hyperplane complements are simultaneously more and less complicated. In this article, we show not only that the cohomology ring of the complement is the same algebra as in the complex case, up to a multiplication of indices by $3$, but that the rational homotopy type can be determined entirely by the cohomology ring.