The structure theory of Nilspaces III: Inverse limit representations and topological dynamics

This paper forms the third part of a series by the authors [GMV1,GMV2] concerning the structure theory of nilspaces. A nilspace is a compact space $X$ together with closed collections of cubes $C^n(X)\subseteq X^{2^n}$, $n=1,2,\ldots$, satisfying some natural axioms. Our goal is to extend the structure theory of nilspaces obtained by Antol\'in Camarena and Szegedy, and to provide new proofs. Our main result is that, under the technical assumption that $C^n(X)$ is a connected space for all $n$, then $X$ is isomorphic (in a strong sense) to an inverse limit of nilmanifolds. This is a direct and slight generalization of the main result of Antol\'in Camarena and Szegedy. We also apply our methods to obtain structure theorems in the setting of topological dynamics. Specifically, if $H$ is a group (subject to very mild topological assumptions) and $(H,X)$ is a minimal dynamical system, then we give a simple characterization of the maximal pronilfactor of $X$. This generalizes the case $H = \mathbb{Z}$, which is a theorem of Host, Kra and Maass, although even in that case we give a significantly different proof.