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Publication# Lq,p-cohomology of Riemannian manifolds and simplicial complexes of bounded geometry

Abstract

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.

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Geometry

Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

Cohomology

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

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