Mapping class groups of exotic tori and actions by ${\mathrm{SL}}_d(\mathbb{Z})$

Abstract

We determine for which exotic tori $\mathcal{T}$ of dimension $d\ne 4$ the homomorphism from the group of isotopy classes of orientation-preserving diffeomorphisms of $\mathcal{T}$ to ${\mathrm{SL}}_d(\mathbb{Z})$ given by the action on the first homology group is split surjective. As part of the proof we compute the mapping class group of all exotic tori $\mathcal{T}$ that are obtained from the standard torus by a connected sum with an exotic sphere. Moreover, we show that any nontrivial ${\mathrm{SL}}_d(\mathbb{Z})$-action on $\mathcal{T}$ agrees on homology with the standard action, up to an automorphism of ${\mathrm{SL}}_d(\mathbb{Z})$. When combined, these results in particular show that many exotic tori do not admit any nontrivial differentiable action by ${\mathrm{SL}}_d(\mathbb{Z})$.

arXiv comment26 pages, to appear in Transactions of the American Mathematical Society