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

TL;DR

This paper proves that there is no $C_2$-ring spectrum combining spin$^c$ bordism and spin bordism in a way that aligns with their fixed point spectra, highlighting a fundamental incompatibility.

Contribution

It establishes a nonexistence result for a specific $C_2$-ring spectrum connecting spin$^c$ and spin bordism theories.

Findings

No $C_2$-ring spectrum with underlying $ ext{MSpin}^c$ and fixed points $ ext{MSpin}$ exists.

Clarifies limitations in constructing equivariant refinements of bordism theories.

Provides insight into the structure of equivariant ring spectra in algebraic topology.

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 $\mathrm{MSpin}$.