Minimal polynomial of the stable-cohomotopy transfer class

Determine the minimal polynomial of the stable-cohomotopy class \(\kappa_0^{S}\in \pi_S^0(\mathcal{M}_g)\) associated to the universal genus-\(g\) surface bundle, and clarify the meaning of this minimal polynomial.

Background

The paper proves that a monic polynomial constructed from Euler characteristics of fixed-point surfaces has a power that annihilates κ0S\kappa_0^S in stable cohomotopy. This establishes an annihilating relation up to nilpotence but does not determine the minimal polynomial.

For genera 2 and 3, explicit annihilating polynomials are computed from classifications of finite subgroups of the mapping class groups. The example then asks for the minimal polynomial of the corresponding class in a generalised cohomology theory EE, together with an interpretation of what such a polynomial would signify.

References

How can one determine the minimal polynomial of \kappa_0{E} \in E0(\mathcal{M}_g)? What does it mean?

— Mumford--Morita--Miller classes in generalised cohomology theories  (2610.01888 - Randal-Williams, 1 Oct 2026) in Section 8, “The class \(\kappa_0\),” Example immediately following the theorem on polynomial annihilation of \(\kappa_0^S\)