On the Cohomology of the Mapping Class Group of the Punctured Projective Plane

The mapping class group $\Gamma ^k(N_g)$ of a non-orientable surface with punctures is studied via classical homotopy theory of configuration spaces. In particular, we obtain a non-orientable version of the Birman exact sequence. In the case of ${\mathbb{R}} \textrm{P}^2$, we analyze the Serre spectral sequence of a fiber bundle $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k \to X_k \to BSO(3)$ where $X_k$ is a $K(\Gamma ^k({\mathbb{R}} \textrm{P}^2),1)$ and $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k$ denotes the configuration space of unordered $k$-tuples of distinct points in ${\mathbb{R}} \textrm{P}^2$. As a consequence, we express the mod-2 cohomology of $\Gamma ^k({\mathbb{R}} \textrm{P}^2)$ in terms of that of $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k$.

Homology of Borel Subgroup of SL(2,\mathbb{F}_p)

In this paper we compute the integral homology of the Borel subgroup $B$ of the special linear group $SL(2,\mathbb{F}_p), p$ is a prime number. Arcoding to Adem \cite{AJM} these are periodic groups. In order to compute the integral homology of $B,$ we decompose it into $\ell-$ primary parts. We compute the first summand based on Invariant Theory and compute the rest summand based on Lyndon-Hochschild-Serre spectral sequence. We assume that $p$ is an odd prime and larger than $3.$

Cohomology of Groups

This chapter introduces the basic ingredients of the cohomology of groups and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: integral homology of finite groups such as the Mathieu groups, homology of crystallographic groups, homology of nilpotent groups, homology of Coxeter groups, transfer homomorphism, homological perturbation theory, mod-p comology rings of small finite p-groups, Lyndon-Hocshild-Serre spectral sequence, Bokstein operation, Steenrod squares, Stiefel-Whitney classes, Lie algebras, the modular isomorphism problem, and Bredon homology.

On the Cohomology of the Classifying Spaces of Projective Unitary Groups

Let [Formula: see text] be the classifying space of [Formula: see text], the projective unitary group of order [Formula: see text], for [Formula: see text]. We use a Serre spectral sequence to determine the ring structure of [Formula: see text] up to degree [Formula: see text], as well as a family of distinguished elements of [Formula: see text], for each prime divisor [Formula: see text] of [Formula: see text]. We also study the primitive elements of [Formula: see text] as a comodule over [Formula: see text], where the comodule structure is given by an action of [Formula: see text] on [Formula: see text] corresponding to the action of taking the tensor product of a complex line bundle and an [Formula: see text]-dimensional complex vector bundle.

Nonabelian holomorphic Lie algebroid extensions

We classify nonabelian extensions of Lie algebroids in the holomorphic category. Moreover we study a spectral sequence associated to any such extension. This spectral sequence generalizes the Hochschild–Serre spectral sequence for Lie algebras to the holomorphic Lie algebroid setting. As an application, we show that the hypercohomology of the Atiyah algebroid of a line bundle has a natural Hodge structure.

A LERAY SPECTRAL SEQUENCE FOR NONCOMMUTATIVE DIFFERENTIAL FIBRATIONS

This paper describes the Leray spectral sequence associated to a differential fibration. The differential fibration is described by base and total differential graded algebras. The cohomology used is noncommutative differential sheaf cohomology. For this purpose, a sheaf over an algebra is a left module with zero curvature covariant derivative. As a special case, we can recover the Serre spectral sequence for a noncommutative fibration.

EXT ALGEBRA OF NICHOLS ALGEBRAS OF CARTAN TYPE A2

We give the full structure of the Ext algebra of any Nichols algebra of Cartan type A2 by using the Hochschild–Serre spectral sequence. As an application, we show that the pointed Hopf algebras [Formula: see text] with Dynkin diagrams of type A, D, or E, except for A1 and A1 × A1 with the order NJ > 2 for at least one component J, are wild.