Mapping class groups of exotic tori and actions by 𝑆𝐿_{𝑑}(𝐙)

We determine for which exotic tori $\tau$ of dimension d $\neq$ 4 the homomorphism from the group of isotopy classes of orientation-preserving diffeomorphisms of $\tau$ to SL$_d$(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 $\tau$ that are obtained from the standard torus by a connected sum with an exotic sphere. Moreover, we show that any nontrivial SL$_d$(Z)-action on $\tau$ agrees on homology with the standard action, up to an automorphism of SL$_d$(Z). When combined, these results in particular show that many exotic tori do not admit any nontrivial differentiable action by SL$_d$(Z).