Chern class obstructions to smooth equivariant rigidity

Abstract

By work of Kirby-Siebenmann \cite{KirbySiebenmann} and Kervaire-Milnor \cite{KervaireMilnor}, there are only finitely many smooth manifolds homeomorphic to a given closed topological manifold. A construction involving Whitehead torsion shows this is not the case equivariantly for smooth finite group actions on a product $M\times I$ (see \cite[p. 262-266]{BrowderHsiangProblem}). When $2$ has odd order in $\left(\mathbb{Z}/p\mathbb{Z}\right)^\times$, Schultz \cite{SchultzSpherelike} uses a different method involving the Atiyah-Singer index theorem and computations of Ewing \cite{EwingSpheresAsFPSets} to show that there are infinitely many equivariant smooth structures for certain actions of $G=\mathbb{Z}/p\mathbb{Z}$ on even dimensional spheres with fixed point set $S^2$. These examples are constructed by finding infinitely many $G$-vector bundles over $S^2$ with vanishing Atiyah-Singer class and using these vector bundles to replace the normal bundle of $S^2\subseteq S^{2n}$. We analyze when a manifold supports infinitely many $G$-vector bundles with vanishing Atiyah-Singer class and show that Schultz's examples of exotic equivariant manifolds can be extended to much greater generality. As a consequence, we see that, for infinitely many primes $p$, there are infinitely many stable $G$-smoothings of a smooth $G$-manifold in the sense of Lashof \cite{LashofStableGSmoothing} whenever the fixed set has nonzero second rational cohomology.