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The structure theory of nilspaces I
- Source :
- Journal d'Analyse Mathématique, vol 140, iss 1
- Publication Year :
- 2020
- Publisher :
- eScholarship, University of California, 2020.
-
Abstract
- This paper forms the first part of a series by the authors [GMV2,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. Antol\'in 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 of maps that we term fibrations, which are essentially equivalent to what are termed fiber-surjective morphisms by Anatol\'in Camarena and Szegedy, and we 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.<br />Comment: 64 pages, minor revision based on referee's report, results and proofs have not been changed, this version is accepted for publication in J. Anal. Math
- Subjects :
- medicine.medical_specialty
Class (set theory)
Pure mathematics
General Mathematics
Topological dynamics
Dynamical Systems (math.DS)
01 natural sciences
Morphism
math.GN
0103 physical sciences
medicine
FOS: Mathematics
Mathematics - Combinatorics
math.CO
Mathematics - Dynamical Systems
0101 mathematics
Abelian group
Axiom
Mathematics - General Topology
Mathematics
010102 general mathematics
General Topology (math.GN)
Pure Mathematics
Compact space
010307 mathematical physics
Inverse limit
Combinatorics (math.CO)
math.DS
Analysis
Structured program theorem
Subjects
Details
- Database :
- OpenAIRE
- Journal :
- Journal d'Analyse Mathématique, vol 140, iss 1
- Accession number :
- edsair.doi.dedup.....d51e82a1645dfb0e47f1e84fb9124646