The Rational Homotopy of Stable $C_p$-Smoothings

Smooth structures on high dimensional manifolds are classified by maps to the infinite loop space $TOP\mathrm{/}O$. The homotopy groups of this space are known to be finite. Given a compact Lie group $G$, this space can be regarded as an equivariant infinite loop space and equivariant maps from a locally linear, high dimensional $G$-manifold to $TOP\mathrm{/}O$ classify stable $G$-smoothings. We compute the equivariant homotopy groups ${\pi }_V^{C_p}TOP\mathrm{/}O\otimes \mathbb{Q}$ where $C_p$ denotes the cyclic group of order $p$. By applying our methods to the group $C_4$, we prove a Chern class analogue of Novikov’s theorem on rational Pontryagin classes.