Edge invariant for 3D interacting Floquet systems with translation symmetry

In two dimensions, interacting Floquet topological phases may arise even in the absence of protecting symmetry. These phases are classified by the chiral edge transport at the one dimensional boundary of an open system, which is captured by a chiral unitary index of the effective one dimensional unitary. We study an extension of this index to two dimensional unitaries with translation symmetry. We use this extended index to work towards a classification of 3D interacting topological phases with no symmetry except translation invariance.