Simplicial homology

In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case of dimension 0). Simplicial homology arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles. This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex). By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a triangulation of the given space. Many topological spaces of interest can be triangulated, including every smooth manifold (Cairns and Whitehead). Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology onl

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Cellular Homotopy Excision

Kay Remo Werndli

There is a classical "duality" between homotopy and homology groups in that homotopy groups are compatible with homotopy pullbacks (every homotopy pullback gives rise to a long exact sequence in homotopy), while homology groups are compatible with homotopy pushouts (every homotopy pushout gives rise to a long exact sequence in homology). This last statement is sometimes referred to as the Mayer-Vietoris or excision axiom. The classical Blakers-Massey theorem (or homotopy excision theorem) asks to what extent the excision property for homotopy pushouts remains true if we replace homology groups by homotopy groups and gives a range in which the excision property holds. It does so by estimating the connectivity of a certain comparison map, which is a rather crude measure, as it is just a single number. Since connectivity is a special case of a cellular inequality, the hope is that there is a stronger statement hidden behind the connectivity result in terms of such inequalities. This process of generalising the homotopy excision theorem has been initiated by Chachólski in the 90s, where he proved a more general version for homotopy pushout squares. The caveat was that one had to suspend the comparison map in question first and the goal of our project -- which we obtained -- was to lose this suspension and then move on to cubical diagrams, rather than squares. To do so, there are a few basic ingredients that are necessary. We first talk about our abstract approach to derived functors, then construct left Bousfield localisations of combinatorial model categories and finally, generalise the foundational concepts in the theory of closed classes to non-connected spaces.

EPFL2016

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High order discontinuous Galerkin methods on simplicial elements for the elastodynamics equation

Carlo Marcati, Alfio Quarteroni

In this work we apply the discontinuous Galekin (dG) spectral element method on meshes made of simplicial elements for the approximation of the elastodynamics equation. Our approach combines the high accuracy of spectral methods, the geometrical flexibility of simplicial elements and the computational efficiency of dG methods. We analyze the dissipation, dispersion and stability properties of the resulting scheme, with a focus on the choice of different sets of basis functions. Finally, we apply the method on benchmark as well as realistic test cases.

Springer Verlag2016

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

Stephen Ducret

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.

EPFL2009

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Homologie (mathématiques)

En mathématiques, l'homologie est une manière générale d'associer une séquence d'objets algébriques tels que des groupes abéliens ou des modules à d'autres objets mathématiques tels que des espaces t

Topologie algébrique

La topologie algébrique, anciennement appelée topologie combinatoire, est la branche des mathématiques appliquant les outils de l'algèbre dans l'étude des espaces topologiques. Plus exactement, elle

Complexe différentiel

En mathématiques, un complexe différentiel est un groupe abélien (voire un module), ou plus généralement un objet d'une catégorie abélienne, muni d'un endomorphisme de carré nul (appelé différentiell

MATH-323: Algebraic topology

Homology is one of the most important tools to study topological spaces and it plays an important role in many fields of mathematics. The aim of this course is to introduce this notion, understand its properties and learn how to compute it. There will be many examples and applications.

MATH-410: Riemann surfaces

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex domains under discontinuous group actions, as algebraic curves.

MATH-436: Homotopical algebra

This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous examples of model categories and their applications in algebra and topology.