Jorina Schuett:Introduction to Differential Manifolds and their Cohomology

This exposition starts with basic algebraic definitions such as module, tensor product, tensor algebra, as well as symmetric and skew-symmetric algebras. Next, we define n-dimensional manifolds as topological spaces locally homeomorphic to balls in ℝ^n. In the smooth case we define differential forms of arbitrary orders. Then we will prove the fundamental Stokes theorem for differential forms, which, in particular, explain how a surface integral of a vector field over an oriented surface is related to the volume integral of its divergence over the body bounded by the surface. We will investigate Stokes theorem for cuboids, simplices and general manifolds. Finally, we define the notion of de Rham cohomology of a smooth manifold using the famous Poincaré lemma. De Rham cohomology is a analytical way of approaching the algebraic topology of a manifold. De Rham theorem claims that de Rham cohomology group of a manifold is isomorphic with its singular homology group. For purposes of illustration, we provide a connection between the vector analytic notions such as gradient, divergence and curl in ℝ³ and the singular homology of the corresponding objects. At the end we will take a look at Morse inequalities. Morse theory gives a direct way of analyzing the topology of a manifold by studying smooth real-valued functions on it.