Assuming the fundamental group has been developed we will see how this piece of algebraic topology controls two dimensional manifolds and three dimensional manifolds. More algebraic topology is needed to begin to treat four and higher dimensional manifolds...higher homotopy groups and homology groups.
One way to understand the homology groups of a space is as a functor on the homotopy category of cell complexes satisfying exactness and excision. Without assuming a normalising dimension axiom this allows other functors like K theory and bordism theory as well. All of these may be described covariantly by geometric cycles mapping into the target spaces. They may also be described contravariantly by geometric objects over base spaces like vector bundles or block bundles.
We will discuss how differential forms give a purely algebraic model of the homotopy category. And how differential forms provide a representation of the real characteristic classes of vector bundles.
Some of the planned discussion may only appear in the spring semester ... like the refinement of the Chern Weil characteristic differential forms of vector bundles with connection and the integral characteristic classes of vector bundles provided by the Cheeger Simons differential characters of vector bundles with connection.