Homologie (mathématiques)

Résumé

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 topologiques. Les groupes d'homologie ont été définis à l'origine dans la topologie algébrique. Des constructions similaires sont disponibles dans beaucoup d'autres contextes, tels que l'algèbre abstraite, les groupes, les algèbres de Lie, la théorie de Galois et la géométrie algébrique. La motivation initiale pour définir les groupes d'homologie était l'observation que deux formes peuvent être distinguées en examinant leurs trous. Par exemple, un cercle n'est pas un disque car le cercle est perforé alors que le disque est solide et la sphère n'est pas un cercle car la sphère renferme un trou bidimensionnel alors que le cercle renferme un trou unidimensionnel. Cependant, étant donné qu’un trou n’est "pas là", la définition d'un trou et comment distinguer différents types de trous n'est pas

Source officielle

https://fr.wikipedia.org/wiki/Homologie_(math%C3%A9matiques)

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Torsion homology of arithmetic lattices and K-2 of imaginary fields

Vincent Emery

Let be a symmetric space of noncompact type. A result of Gelander provides exponential upper bounds in terms of the volume for the torsion homology of the noncompact arithmetic locally symmetric spaces . We show that under suitable assumptions on this result can be extended to the case of nonuniform arithmetic lattices that may contain torsion. Using recent work of Calegari and Venkatesh we deduce from this upper bounds (in terms of the discriminant) for of the ring of integers of totally imaginary number fields . More generally, we obtain such bounds for rings of -integers in F.

Springer Heidelberg2014

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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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We first explore the notion of G-manifold, cobordism and then discuss the generalized Pontrjagin-Thom Theorem. We then compute the cobordism rings Ω (...) using stable homotopy theory. Secondly, we investigate the relations between cohomology theories and formal group laws. This is the very beginning of the so called chromatic homotopy theory. For any complex oriented cohomology theories, we associate a formal group law. We show that the formal group law associated to complex cobordism is universal. Finally, for a suitable formal group law we give a way to realize it into a generalized homology theory.

2011

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