$\mathrm{Spin}^c$–structures and homotopy equivalences

Abstract

We show that a homotopy equivalence between manifolds induces a correspondence between their spin${}^c$–structures, even in the presence of 2–torsion. This is proved by generalizing spin${}^c$–structures to Poincaré complexes. A procedure is given for explicitly computing the correspondence under reasonable hypotheses.