The structure theory of nilspaces I

Abstract

This paper forms the first part of a series by the authors [GMV16a, GMV16b] concerning the structure theory of nilspaces of Antolín Camarena and Szegedy. A nilspace is a compact space X together with closed collections of cubes Cn(X) ⊑ X2n, n = 1, 2,... satisfying some natural axioms. Antolín Camarena and Szegedy proved that from these axioms it follows that (certain) nilspaces are isomorphic (in a strong sense) to an inverse limit of nilmanifolds. The aim of our project is to provide a new self-contained treatment of this theory and give new applications to topological dynamics.This paper provides an introduction to the project from the point of view of applications to higher order Fourier analysis. We define and explain the basic definitions and constructions related to cubespaces and nilspaces and develop the weak structure theory, which is the first stage of the proof of the main structure theorem for nilspaces. Vaguely speaking, this asserts that a nilspace can be built as a finite tower of extensions where each of the successive fibers is a compact abelian group.We also make some modest innovations and extensions to this theory. In particular, we consider a class ofmaps thatwe term fibrations, which are essentially equivalent to what are termed fiber-surjective morphisms by Anatolín Camarena and Szegedy; andwe formulate and prove a relative analogue of the weak structure theory alluded to above for these maps. These results find applications elsewhere in the project.