Fixed-point description of the fundamental complex-cobordism MMM class

Determine whether the complex-cobordism Mumford–Morita–Miller class \(\kappa_{-1}^{MU}(\rho)\) associated to an arbitrary finite-group action \(\rho\) on an oriented surface admits a description in terms of the action’s fixed-point data.

Background

For a finite group action $\rho\colon G\to \mathrm{Diff}^+(\Sigma)$, the paper derives fixed-point formulas for the complex-cobordism Mumford–Morita–Miller classes $\kappa_i^{MU}(\rho)$ when $i\geq 0$. These formulas express the classes using isotropy groups and tangent representations at points with nontrivial stabiliser.

The class $\kappa_{-1}^{MU}(\rho)=\pi_!^{MU}(1)$ behaves differently from the higher classes. The paper obtains a fixed-point formula for this class in the special case of cyclic group actions, but explicitly leaves unresolved whether an analogous description exists for arbitrary finite groups.