What question did this study set out to answer?

The aim is to demonstrate that a specific C2-ring spectrum does not exist for spin^c bordism.

March 31, 2026Open Access

On the nonexistence of a Green functor with values spinᶜ bordism and spin bordism

Key Points

The aim is to demonstrate that a specific C2-ring spectrum does not exist for spin^c bordism.
Analyzed the properties of C2-ring spectra and their fixed point spectra.
Examined the relationships between MSpin and MSpin^c in context of bordism theory.
Applied theoretical proofs to establish the nonexistence of the Green functor.
Proved that no C2-ring spectrum exists with MSpin^c as the underlying ring spectrum.
Demonstrated that Atiyah–Bott–Shapiro orientations cannot be derived from a single C2-ring spectrum.
Clarified the implications of this nonexistence for the study of bordism in algebraic topology.

Abstract

Abstract We show that there does not exist a C₂ C 2 -ring spectrum whose underlying ring spectrum is MSpinᶜ MSpin c and whose C₂ C 2 -fixed point spectrum is MSpin MSpin. As a corollary, the spinᶜ spin c and spin orientations of Atiyah–Bott–Shapiro cannot be obtained as the underlying and fixed point maps of a single map of C₂ C 2 -ring spectra.

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