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

Abstract

In this note, we show that there does not exist a $C_2$-ring spectrum whose underlying ring spectrum is ${\mathrm{MSpin}}^c$ and whose $C_2$-fixed point spectrum is $\mathrm{MSpin}$.

v2: 5 pages, added context to the introduction, added clarifying discussions, and reorganized the statements and proofs of the main results; submitted version