Spectrum-level ring lift of the universal Tutte–Grothendieck invariant

Establish the existence of a map of ring spectra K(Mat~) -> K(Mat_tc) whose induced morphism on K0 equals the universal Tutte–Grothendieck invariant y: Z[M] -> RTG; equivalently, determine whether the spectrum-level lift K(T): K(Mat~) -> K(Mat_tc) can be promoted to a homomorphism of ring spectra once K(Mat~) and K(Mat_tc) are endowed with ring spectrum structures compatible with the direct sum of matroids.

Background

The paper constructs two categories with covering families based on matroids, Mat~ = (Mat+, S~ , *) and Mat_tc = (Mat+, S_tc, *), and shows that the universal Tutte–Grothendieck invariant y: Z[M] -> RTG is realized on K0 by a map of spectra K(T): K(Mat~) -> K(Mat_tc).

While Theorem C lifts y to the spectral level as a map of spectra, it does not establish multiplicative (ring) compatibility at the spectrum level. The authors point to Zakharevich’s framework for symmetric monoidal assemblers, where K-theory acquires an E∞-ring (denoted Ex-ring in the text) structure and functoriality induces maps of ring spectra, suggesting a route to promote K(T) to a ring-spectrum map.

The open question is whether one can equip the categories with covering families Mat~ and Mat_tc with suitable symmetric monoidal structures (arising from direct sum of matroids) so that their K-theory spectra become ring spectra and K(T) is a homomorphism of ring spectra, thereby yielding a spectrum-level lift of the universal Tutte–Grothendieck invariant as a ring homomorphism.