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By work of Kirby-Siebenmann \cite{KirbySiebenmann} and Kervaire-Milnor\cite{KervaireMilnor}, there are only finitely many smooth manifoldshomeomorphic to a given closed topological manifold. A construction involvingWhitehead torsion shows this is not the case equivariantly for smooth finitegroup 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 andcomputations of Ewing \cite{EwingSpheresAsFPSets} to show that there areinfinitely 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 $G$-vectorbundles over $S^2$ with vanishing Atiyah-Singer class and using these vectorbundles to replace the normal bundle of $S^2\subseteq S^{2n}$. We analyze whena manifold supports infinitely many $G$-vector bundles with vanishingAtiyah-Singer class and show that Schultz's examples of exotic equivariantmanifolds can be extended to much greater generality. As a consequence, we seethat, 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 secondrational cohomology.

Chern class obstructions to smooth equivariant rigidity