The Rational Homotopy of Stable $C_p$-Smoothings
Oliver H. Wang
TL;DR
This work computes the rational Cp-equivariant homotopy groups of TOP/O by embedding the problem in G-smoothing theory, reducing to fixed-set data, and leveraging equivariant pro-structures and comparison with L-theory. For odd primes p, the rational Cp-equivariant groups are described via a kernel 𝔈_p of Ewing-type relations, with dim 𝔈_p = (p−1)/(2t) where t is the order of 2 in 𝔽_p^×, and a congruence-driven pattern across degrees; for p=2 all rational Cp-equivariant groups vanish. The approach synthesizes Madsen–Rothenberg equivariant automorphism theory, Ewing's fixed-set relations, and smoothing results of Schultz and Cappell–Weinberger to identify rational lifts and compute π_* of TOP_{C_p}/O_{C_p}. The paper also proves a fixed-set reduction [X, TOP/O]^{C_p} ≃_ℚ [X^{C_p}, TOP_{C_p}/O_{C_p}] and discusses the topological invariance of rational Chern and Pontryagin classes in the equivariant setting via stabilization and Atiyah–Singer G-signature. Together, these results illuminate how stable G-smoothings behave rationally and connect smoothing theory with equivariant K- and L-theory invariants.
Abstract
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 $π_V^{C_p}TOP/O\otimes\Q$ where $C_p$ denotes the cyclic group of order $p$.