Nilspaces, nilmanifolds and their morphisms
Omar Antolin Camarena, Balazs Szegedy

TL;DR
This paper introduces nilspaces, a new algebraic structure generalizing nilmanifolds, and explores their properties and connections to ergodic theory, additive combinatorics, and higher order Fourier analysis.
Contribution
It develops the theory of nilspaces, showing they are inverse limits of finite dimensional structures and that finite dimensional connected nilspaces are nilmanifolds.
Findings
Compact nilspaces are inverse limits of finite dimensional ones
Finite dimensional connected nilspaces are nilmanifolds
Nilspaces generalize the theory of compact abelian groups
Abstract
Recent developments in ergodic theory, additive combinatorics, higher order Fourier analysis and number theory give a central role to a class of algebraic structures called nilmanifolds. In the present paper we continue a program started by Host and Kra. We introduce nilspaces as structures satisfying a variant of the Host-Kra axiom system for parallelepiped structures. We give a detailed structural analysis of abstract and compact topological nilspaces. Among various results it will be proved that compact nilspaces are inverse limits of finite dimensional ones. Then we show that finite dimensional compact connected nilspaces are nilmanifolds. The theory of compact nilspaces is a generalization of the theory of compact abelian groups. This paper is the main algebraic tool in the second authors approach to Gowers's uniformity norms and higher order Fourier analysis.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Mathematical Dynamics and Fractals
