  We present a detailed path integral derivation of the topological response of
gapped free fermions, in $2+1$ dimensions, to an external $\text{U}(1)$ Gauge
field. The well-known Hall response is obtained by identifying the Chern-Simons
term in the effective action with the correct coefficient. We extend the result
to $2d+1$ dimensions in which the response is associated to a Chern-Simons term
with a coefficient related to a characteristic class coming from topological
band theory. We comment on the bulk-to-boundary principle which arises
naturally when one considers the theory on a manifold with boundary.
