The Cohomology of Quaternionic Hyperplane Complements

TL;DR

This paper investigates the topological and algebraic properties of quaternionic hyperplane complements, revealing that their cohomology ring mirrors the complex case and that their rational homotopy type is fully determined by this cohomology.

Contribution

It establishes that quaternionic hyperplane complements have a cohomology ring similar to the complex case, and their rational homotopy type can be derived solely from this ring.

Findings

Cohomology ring of quaternionic hyperplane complements matches the complex case up to index multiplication by 3.

Rational homotopy type is completely determined by the cohomology ring.

Hyperplane complements over quaternions are simply connected, contrasting with the complex case.

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.