On the nonexistence of a Green functor with values $$\hbox {spin}^c$$ bordism and spin bordism

We show that there does not exist a $$C_2$$ -ring spectrum whose underlying ring spectrum is $$\textrm{MSpin}^c$$ and whose $$C_2$$ -fixed point spectrum is $$\textrm{MSpin}$$ . As a corollary, the $$\hbox {spin}^c$$ and spin orientations of Atiyah–Bott–Shapiro cannot be obtained as the underlying and fixed point maps of a single map of $$C_2$$ -ring spectra.