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Computation of refined TC- for MU over Q

Compute the even homotopy groups (and associated algebraic description) of the refined invariant −,((MU⊗Q)/MU), i.e., determine π2∗−,((MU⊗Q)/MU), analogous to the ku and KU cases analyzed in this paper.

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Background

In discussing how to approach (THHref(Q)), the authors suggest base changing to MU and then using Adams–Novikov descent. They fully compute the ku and KU cases in this work but leave the MU case unresolved.

A computation for MU would be a significant step toward understanding the refined invariant in higher chromatic settings and could inform potential extensions to BP⟨n⟩ or En.

References

While we still don't know what happens for MU, the purpose of this article is to give an answer for ku.

— Derived $q$-Hodge complexes and refined $\operatorname{TC}^-$ (2410.23115 - Meyer et al., 30 Oct 2024) in Section 1.3 (What’s THHref(Q)?), paragraph introducing the MU base-change strategy