Geometric construction of the quantum Hall effect in all even dimensions

The quantum Hall effects in all even dimensions are uniformly constructed. Contrary to some recent accounts in the literature, the existence of quantum Hall effects (QHE) does not crucially depend on the existence of division algebras. For QHE on flat space of even dimensions, both the Hamiltonians and the ground-state wavefunctions for a single particle are explicitly described. This explicit description immediately tells us that QHE on a higher even-dimensional flat space shares common features such as incompressibility with QHE on a plane.