Symmetries of exotic aspherical space forms

2021/09/19 by Mauricio Bustamante, Bena Tshishiku, Bustamante, Mauricio +1
Mathematics · #22F30 (secondary) #57R55 #57R60 #57S17 (primary) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2109.09196

openalex publication_date 2021/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study finite group actions on smooth manifolds of the form M#Σ, where Σ is an exotic n-sphere and M is a closed aspherical space form. We give a classification result for free actions of finite groups on M#Σ when M is 7-dimensional. We show that if \mathbb Z/p\mathbb Z acts freely on Tⁿ#Σ, then Σ is divisible by p in the group of homotopy spheres. When M is hyperbolic, we give examples M#Σ that admit no nontrivial smooth action of a finite group, even though Isom(M) is arbitrarily large. Our proofs combine geometric and topological rigidity results with smoothing theory and computations with the Atiyah--Hirzebruch spectral sequence.

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