  In this paper, we derive the explicit formula for higher cup products on
hypercubic lattices, based on the recently developed geometrical interpretation
in the simplicial case. We illustrate how this formalism can elucidate lattice
constructions on hypercubic lattices for various models and deriving them from
spacetime actions. In particular, we demonstrate explicitly that the (3+1)D SPT
$S = \frac{1}{2} \int w_2^2 + w_1^4$ (where $w_1$ and $w_2$ are the first and
second Stiefel-Whitney classes) is dual to the 3-fermion Walker-Wang model
constructed on the cubic lattice by Burnell-Chen-Fidkowski-Vishwanath. Other
examples include the double-semion model, and also the `fermionic' toric code
in arbitrary dimensions on hypercubic lattices. In addition, we extend previous
constructions of exact boson-fermion dualities and the Gu-Wen Grassmann
Integral to arbitrary dimensions. Another result which may be of independent
interest is a derivation of a cochain-level action for the generalized
double-semion model, reproducing a recently derived action on the cohomology
level.
