Smooth structures on high dimensional manifolds are classified by maps to the infinite loop space $TOP/O$. 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 $TOP/O$ classify stable $G$-smoothings. We compute the equivariant homotopy groups $\pi_V^{ C_p}TOP/O\otimes\Q$ where $C_p$ denotes the cyclic group of order $p$.
Categories: math.AT math.GT