The structure theory of Nilspaces II: Representation as nilmanifolds

TL;DR

This paper proves that certain structured spaces called nilspaces, when their fibers are tori, are equivalent to nilmanifolds, and extends this to dynamical systems, advancing the structure theory of nilspaces.

Contribution

Provides a new proof that nilspaces with torus fibers are isomorphic to nilmanifolds and extends the result to dynamical systems, enhancing the understanding of nilspace structures.

Findings

Nilspaces with torus fibers are isomorphic to nilmanifolds.

Extension of the theorem to nilspaces arising from dynamical systems.

Development of solutions for functional equations to extract algebraic structure.

Abstract

This paper forms the second part of a series by the authors [GMV1,GMV3] concerning the structure theory of nilspaces of Antol\'in Camarena and Szegedy. 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. From these axioms it follows that a nilspace can be built as a finite tower of extensions where each of the successive fibers is a compact abelian group. Our main result is a new proof of a result due to Antol\'in Camarena and Szegedy [CS12], stating that if each of these groups is a torus then $X$ is isomorphic (in a strong sense) to a nilmanifold $G/\Gamma$. We also extend the theorem to a setting where the nilspace arises from a dynamical system $(X,T)$. These theorems are a key stepping stone towards the general structure theorem in [GMV3] (which again closely resembles the main theorem of [CS12]). The main technical tool, enabling us to deduce algebraic information from topological data, consists of existence and uniqueness results for solutions of certain natural functional equations, again modelled on the theory in [CS12].