  We extend the previous results of exact bosonization, mapping from fermionic
operators to Pauli matrices, in 2d and 3d to arbitrary dimensions. This
bosonization map gives a duality between any fermionic system in arbitrary $n$
spatial dimensions and a new class of $(n-1)$-form $\mathbb{Z}_2$ gauge
theories in $n$ dimensions with a modified Gauss's law. This map preserves
locality and has an explicit dependence on the second Stiefel-Whitney class and
a choice of spin structure on the manifold. A new formula for Stiefel-Whitney
homology classes on lattices is derived. In the Euclidean path integral, this
exact bosonization map is equivalent to introducing a topological "Steenrod
square" term to the spacetime action.
