A smooth Birman-Hilden theory for hyperk\"ahler manifolds

This article grew out of an effort to understand the smooth mapping class groups of certain 4-manifolds in a geometric manner. We prove a smooth analog of the Birman-Hilden theorem for manifolds that admit a hyperk\"ahler structure. This allows us to probe the smooth mapping class groups associated to certain manifolds with nontrivial fundamental groups. Along the way, and of independent interest, we prove a global Torelli theorem for generalized Enriques manifolds. The techniques used are analogous to Teichm\"uller-theoretic methods in the classical theory of mapping class groups. We apply this hyperk\"ahler Birman-Hilden theorem to obtain results regarding smooth, metric, and complex Nielsen realization on Enriques surfaces.