We give examples of Anosov actions of Z² on the d-torus Tᵈ that cannot be homotoped to a smooth action on Tᵈ#Σ, for certain exotic d-spheres Σ. This is deduced using work of Krannich, Kupers, and the authors that, in particular, computes the mapping class group of Tᵈ#Σ.
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Bustamante et al. (2024) studied this question.
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