dc.contributor.author Gutman, Y dc.contributor.author Manners, F dc.contributor.author Varjú, PP dc.date.accessioned 2018-09-05T12:48:08Z dc.date.available 2018-09-05T12:48:08Z dc.date.issued 2020-03 dc.identifier.issn 0021-7670 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/279558 dc.description.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. dc.description.sponsorship Royal Society dc.publisher Springer Science and Business Media LLC dc.title The structure theory of nilspaces I dc.type Article prism.endingPage 369 prism.issueIdentifier 1 prism.publicationDate 2020 prism.publicationName Journal d'Analyse Mathematique prism.startingPage 299 prism.volume 140 dc.identifier.doi 10.17863/CAM.26930 dcterms.dateAccepted 2018-04-29 rioxxterms.versionofrecord 10.1007/s11854-020-0093-8 rioxxterms.licenseref.uri http://www.rioxx.net/licenses/all-rights-reserved rioxxterms.licenseref.startdate 2020-03-01 dc.identifier.eissn 1565-8538 rioxxterms.type Journal Article/Review pubs.funder-project-id Royal Society (UF140146) cam.issuedOnline 2020-04-20 rioxxterms.freetoread.startdate 2019-08-21
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