THE MAYER VIETORIS SEQUENCE CALCULATING DE RHAM COHOMOLOGY
Journal Title: JOURNAL OF ADVANCES IN MATHEMATICS - Year 2016, Vol 12, Issue 8
Abstract
De Rham cohomology it is very obvious that it relies heavily on both topology as well as analysis. We can say it creates a natural bridge between the two. To understand and be able to explain what exactly de Rham cohomology is to the world of mathematics we need to know de Rham groups. This is the reasons to calculate the de Rham cohomology of a manifold. This is usually quite difficult to do directly. We work with manifold. Manifold is a generalization of curves and surfaces to arbitrary dimension. A topological space M is called a manifold of dimension k if : · M is a topological Hausdorff space . · M has a countable topological base. · For all m∈M there is an open neighborhood U⊂M such that U is homeomorphic to an open subset V of â„k. There are many different kinds of manifolds like topological manifolds, â„‚k - manifolds, analytic manifolds, and complex manifolds, we concerned in smooth manifolds. A smooth manifold can described as a topological space that is locally like the Euclidian space of a dimension known. An important definion is homeomorphism. Let X, Y be topological spaces, and let f: X⟶Y e a bijection. If both f and the inverse function f−1: X⟶Y are continuous, then f is called a homeomorphism We introduce one of the useful tools for this calculating, the Mayer – Vietoris sequence. Another tool is the homotopy axiom. In this material I try to explain the Mayer – Vietoris sequence and give same examples. A short exact sequence of cochain complexes gives rise to a long exact sequence in cohomology, called the Mayer - Vietories sequence. Cohomology of the circle (S^1), cohomology of the spheres (S^2). Homeomorphism between vector spaces and an open cover of a manifold. we define de Rham cohomology and compute a few examples.
Authors and Affiliations
Arben Baushi
Method for Optimizing the Dual of Linear Fuzzy Programming Problems
This article’s goal is to support the existence of the dual in a Linear Fuzzy Real environment and focus on its application to Linear fuzzy program problems. This concept will apply to linear fuzzy programming problems...
Closed braids and knot holders associated to some laser dynamical systems: A pump-modulated Nd-doped Â…ber laser
In this work the arising knots and links for the pump-modulated Nd-doped fiber laser is investigated. For the associated templates, some of their topological invariants, such as braid linking matrix, braid words, crossin...
Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method
We present a study on numerical solutions of nonlinear ordinary differential equations by applying Runge-Kutta-Fehlberg (RKF) method, a well-known adaptive Runge-kutta method. The adaptive Runge-kutta methods use embedde...
Martingales via statistical convergence
In this paper martingales of statistical Bochner integrable functions with values in a Banach space are treated. In particular we have arrived some results for martingales and backwards martingales.
Boundedness of the gradient of a solution for the fourth order equation in general domains
Based on new integral estimate, we establish boundedness of the gradient of a solution for a fourth order equation in an arbitrary three-dimensional domain.