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dc.contributor.authorGutman, Y
dc.contributor.authorManners, F
dc.contributor.authorVarjú, PP
dc.date.accessioned2018-09-05T12:48:08Z
dc.date.available2018-09-05T12:48:08Z
dc.date.issued2020-03
dc.identifier.issn0021-7670
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/279558
dc.description.abstractThis 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.sponsorshipRoyal Society
dc.publisherSpringer Science and Business Media LLC
dc.titleThe structure theory of nilspaces I
dc.typeArticle
prism.endingPage369
prism.issueIdentifier1
prism.publicationDate2020
prism.publicationNameJournal d'Analyse Mathematique
prism.startingPage299
prism.volume140
dc.identifier.doi10.17863/CAM.26930
dcterms.dateAccepted2018-04-29
rioxxterms.versionofrecord10.1007/s11854-020-0093-8
rioxxterms.licenseref.urihttp://www.rioxx.net/licenses/all-rights-reserved
rioxxterms.licenseref.startdate2020-03-01
dc.identifier.eissn1565-8538
rioxxterms.typeJournal Article/Review
pubs.funder-project-idRoyal Society (UF140146)
cam.issuedOnline2020-04-20
rioxxterms.freetoread.startdate2019-08-21


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