The central problem in topology is to decide if two spaces are homeomorphic or if two manifolds are diffeomorphic. Algebraic topology offers a possible solution by transforming the geometric problem into an algebraic one. To accomplish this, one associates to each topological space an algebraic object such as its cohomology vector space so that homeomorphic topological spaces correspond to isomorphic vector spaces. Such an association is called a functor. The cohomology functor greatly simplifies the original problem, for the dimension alone determines the isomorphism class of a finite-dimensional vector space. This gives a simple necessary condition for two topological spaces to be homeomorphic.
The guiding principle in this course is to use differential forms as an aid in exploring some of the less digestible aspects of algebraic topology. We will work primarily but not exclusively with smooth manifolds. This has the advantage of simplifying the theory as well as the constructions; for example, three versions of cohomology— singular, de Rham, and ?ech, coincide on smooth manifolds. It allows us to study topics that are normally relegated to a second course in algebraic topology, such as Poincare duality, the Thom isomorphism, and spectral sequences, without assuming a knowledge of homology.
More specifically, the topics to be covered are the following: de Rham cohomology, cohomology with compact support, Mayer–Vietoris sequence, Poincare lemma on the cohomology of ??, degree of a proper map, Brouwer fixed-point theorem, hairy-ball theorem, Poincare duality for a manifold, Kunneth formula for the cohomology of a product, Thom isomorphism for the cohomology of a vector bundle, Euler class, the nonorientable case, ?ech cohomology, isomorphism between de Rham and ?ech cohomology, presheaves, Hopf index theorem on the zeros of a vector field, spectral sequences, cohomology with integer coefficients, and de Rham’s theorem. These topics correspond to most of the first three chapters of the textbook.