Group cohomology

Summary

In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements of G^n representing n-simplices, topological properties of the space may be computed, such as the set of cohomology groups H^n(G,M). The cohomology groups in turn provide insight into the structure of the group G and G-module M themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well

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https://en.wikipedia.org/wiki/Group_cohomology

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Towards new invariants for principal bundles

Martina Rovelli

We investigate the theory of principal bundles from a homotopical point of view. In the first part of the thesis, we prove a classification of principal bundles over a fixed base space, dual to the well-known classification of bundles with a fixed structure group. This leads to an adjointness property in a homotopical context between the classifying space and the loop space. We then focus on characteristic classes, which are invariants for principal bundles that take values in the cohomology of the base space. Each characteristic class captures different geo- metric features of principal bundles. We propose a uniform treatment to interpret most of known characteristic classes as obstructions to group reduction and to the extension of a universal cocycle. By plugging in the correct parameters, the method recovers several classical theorems. Afterwards, we construct a long exact sequence of abelian groups for any principal bundle. This sequence involves the cohomology of the base space and the group cohomology of the structure group. Moreover the connecting map is deeply related with the characteristic classes of the bundle.

EPFL2017

A Looping-Delooping Adjunction For Topological Spaces

Martina Rovelli

Every principal G-bundle over X is classified up to equivalence by a homotopy class X -> BG, where BG is the classifying space of G. On the other hand, for every nice topological space X Milnor constructed a strict model of its loop space (Omega) over tildeX, that is a group. Moreover, the morphisms of topological groups (Omega) over tildeX -> G generate all the G-bundles over X up to equivalence. In this paper, we show that the relation between Milnor's loop space and the classifying space functor is, in a precise sense, an adjoint pair between based spaces and topological groups in a homotopical context. This proof leads to a classification of principal bundles over a fixed space, that is dual to the classification of bundles with a fixed group. Such a result clarifies the deep relation that exists between the theory of bundles, the classifying space construction and the loop space, which are very important in topological K-theory, group cohomology, and homotopy theory.

Int Press Boston, Inc2017

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