  Smith homomorphisms are maps between bordism groups that change both the
dimension and the tangential structure. We give a completely general account of
Smith homomorphisms, unifying the many examples in the literature. We provide
three definitions of Smith homomorphisms, including as maps of Thom spectra,
and show they are equivalent. Using this, we identify the cofiber of the
spectrum-level Smith map and extend the Smith homomorphism to a long exact
sequence of bordism groups, which is a powerful computation tool. We discuss
several examples of this long exact sequence, relating them to known
constructions such as Wood's and Wall's sequences. Furthermore, taking Anderson
duals yields a long exact sequence of invertible field theories, which has a
rich physical interpretation. We developed the theory in this paper with
applications in mind to symmetry breaking in quantum field theory, which we
study in a companion paper.
