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

Abstract We show that there does not exist a $$C_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -ring spectrum whose underlying ring spectrum is $$\textrm{MSpin}^c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>MSpin</mml:mtext> <mml:mi>c</mml:mi> </mml:msup> </mml:math> and whose $$C_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -fixed point spectrum is $$\textrm{MSpin}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>MSpin</mml:mtext> </mml:math> . As a corollary, the $$\hbox {spin}^c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>spin</mml:mtext> <mml:mi>c</mml:mi> </mml:msup> </mml:math> and spin orientations of Atiyah–Bott–Shapiro cannot be obtained as the underlying and fixed point maps of a single map of $$C_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -ring spectra.