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De Rham and Cech-de Rham Cohomologies of Smooth Foliated Manifolds

  • In order to classify smooth foliated manifolds, which are smooth maifolds equipped with a smooth foliation, we introduce the de Rham cohomologies of smooth foliated manifolds. These cohomologies are build in a similar way as the de Rham cohomologies of smooth manifolds. We develop some tools to compute these cohomologies. For example we proof a Mayer Vietoris theorem for foliated de Rham cohomology and show that these cohomologys are invariant under integrable homotopy. A generalization of a known Künneth formula, which relates the cohomologies of a product foliation with its factors, is discussed. In particular, this envolves a splitting theory of sequences between Frechet spaces and a theory of projective spectrums. We also prove, that the foliated de Rham cohomology is isomorphic to the Cech-de Rham cohomology and the Cech cohomology of leafwise constant functions of an underlying so called good cover.

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Metadaten
Author:Thorben Schlierkamp
URN:urn:nbn:de:hbz:385-1-16353
DOI:https://doi.org/10.25353/ubtr-xxxx-68fa-85dc
Series (Volume no.):ohne Schriftenreihe (0)
Publisher:-
Place of publication:-
Referee:Jochen Wengenroth, Leonhard Frerick
Advisor:Jochen Wengenroth
Document Type:Doctoral Thesis
Language:English
Date of completion:2021/07/16
Date of publication:2021/08/03
Publishing institution:Universität Trier
Granting institution:Universität Trier, Fachbereich 4
Date of final exam:2021/05/10
Release Date:2021/08/03
Tag:Cech cohomology of leafwise constant functions; Cech-de Rham cohomology; de Rham cohomology; foliated manifolds
GND Keyword:Differentialgeometrie
Number of pages:116
Institutes:Fachbereich 4
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):License LogoCC BY-NC: Creative-Commons-Lizenz 4.0 International

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