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Lq,p-cohomology of Riemannian manifolds and simplicial complexes of bounded geometry
The Lq,p-cohomology of a Riemannian manifold (M, g) is defined to be the quotient of closed Lp-forms, modulo the exact forms which are derivatives of Lq-forms, where the measure considered comes from the Riemannian structure. The Lq,p-cohomology of a simplicial complex K is defined to be the quotient of p-summable cocycles of K, modulo the coboundaries of q-summable cocycles. We introduce those two notions together with a variant for coarse cohomology on graphs, and we establish their main properties. We define the categories we work on, i.e. manifolds and simplicial complexes of bounded geometry, and we show how cohomology classes can be represented by smooth forms. The first result of the thesis is a de Rham type theorem: we prove that for an orientable, complete and (non compact) Riemannian manifold with bounded geometry (M, g) together with a triangulation K with bounded geometry, the Lq,p-cohomology of the manifold coincides with the Lq,p-cohomology of the triangulation. This is a generalization of an earlier result from Gol'dshtein, Kuz'minov and Shvedov. The second result is a quasi-isometry invariance one: we prove how this de Rham type isomorphism together with a result in coarse cohomology induces the fact that the Lq,p-cohomology of a Riemannian manifold depends only on its quasi-invariance class. This result was proved in the q = p case by Elek. We establish some consequences, such as monocity results for Lq,p-cohomology, and the quasi-isometry invariance of the existence of Sobolev inequalities.
Note: Thèse Ecole polytechnique fédérale de Lausanne EPFL, no 4544 (2009), Programme doctoral Mathématiques, Faculté des sciences de base SB, Institut de géométrie, algèbre et topologie IGAT (Groupe Troyanov GR-TR). Dir.: Marc Troyanov
Keywords: bounded geometry ; coarse cohomology ; Lq,p-cohomology ; théorème de de Rham ; cohomologie grossière ; géométrie bornée ; de Rham theorem ; quasi-isometry invariance ; cohomologie Lq,p ; invariance sous quasi-isométries
Reference
- Additional information: http://library.epfl.ch/theses/?nr=4544
Record created on 2009-10-15, modified on 2012-05-16