The purpose of this unit is to give a brief survey about the cohomology and the tautological rings of the moduli space of curves. Unfortunately, for the most part the cohomology of Mg remains …

15.2 Cohomology groups and Betti numbers We define the k-th de Rham cohomology group of M, denoted Hk(M), to be

Homology and cohomology has its origins in topology, starting with the work of Riemann (1857), Betti (1871) and Poincar e (1895) on \homology numbers" of manifolds. Although Emmy …

Group cohomology 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 …

The moduli space is then the homotopy quotient of this group action, that is the classifying space BG. The cohomology of BG de nes the group cohomology of G. Depending on the action (all …

Note that there is a group homomorphism s : G → M ⋊ G, g 7→(0, g) such that the composite G s−→ M ⋊G π−→ G is the identity map. This is called a splitting of the extension, and the …

Topology motivates thinking of homology and cohomology as functors of a space, and hence of the group G in our setting. Algebra, on the other hand, suggests that we think of homology and …

R1 j X o is a smooth submanifold W which is di eomorphic to R1 the “moduli space of submanifolds of di eomorphic to W ”.

of Mg,n for large k. Recall Harer’s theorem which states that the moduli space Mg,n has the homotopy type of a finite cell complex of dimension 4g − 4 + n for n > 0. Since the homology …

Cohomology of the Moduli Space of Stable Bundles 1 Overview This article is a short review on the basics of Yang{Mills equations over a curve and the main ingredients of the calculations of …