I will then introduce the concepts of a good cover of a manifold, manifolds of finite type, and orientation, all necessary concepts towards the goal of defining and proving Poincaré duality. I will continue with the definition of differential forms on a differentiable manifold and their corresponding spaces necessary to this analysis. In order to define and prove this duality I will start with the simple definition of the dual space of a vector space, with the definition of a positive definite inner product on a vector space, then define the concept of a manifold. This phenomenon is entitled “Poincaré duality” and it describes a general occurrence in differential topology, a duality between spaces of closed, exact differentiable forms on a manifold and their compactly supported counterparts.
In the Poincaré case, lines are given by diameters of the circle or arcs.In this talk I will explain the duality between the deRham cohomology of a manifold M and the compactly supported cohomology on the same space. Remember that in the half-plane case, the lines were either Euclidean lines, perpendicular onto the real line, or half-circles, also perpendicular onto the real line. Moreover, every such intersection is a hyperbolic line. With a circle in the extended complex plane perpendicular to the unit circle bounding Just like in the half-plane model, we will look first at lines in this model. Note that we are still in the complex plane. The underlying space of this model is the open unit disk As you will see at the end of this page, it is this model that appears in Escher's work. You may ask yourself why is it necessary to do that? The reason is that this model is more useful, as it gives more insight and we can make a better use of the apparatus from the complex analysis. This model is constructed starting from the previous one. The second model that we use to represent the hyperbolic plane is called the Poincaré disk model, named after the great French mathematician, Henri Poincaré (1854 - 1912). The Poincaré disk model for the hyperbolic plane The Poincaré model for the hyperbolic plane, Section 7ħ.
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