2023/10/13 by Oliver H. Wang, Wang, Oliver H. · 1 citation
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2310.09363
openalex publication_date 2023/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
By work of Kirby-Siebenmann \citeKirbySiebenmann and Kervaire-Milnor \citeKervaireMilnor, 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× I (see \cite[p. 262-266]BrowderHsiangProblem). When 2 has odd order in (ℤ/pℤ)^×, Schultz \citeSchultzSpherelike uses a different method involving the Atiyah-Singer index theorem and computations of Ewing \citeEwingSpheresAsFPSets to show that there are infinitely many equivariant smooth structures for certain actions of G=ℤ/pℤ on even dimensional spheres with fixed point set S2. These examples are constructed by finding infinitely many G-vector bundles over S2 with vanishing Atiyah-Singer class and using these vector bundles to replace the normal bundle of S2⊆ S2n. 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 \citeLashofStableGSmoothing whenever the fixed set has nonzero second rational cohomology.