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