Exact bosonization in arbitrary dimensions

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 class of (n−1)-form 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 spatial manifold. A 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 space-time action.