In the fall semester (MAT 540), homology theory was treated as part of Geometry in terms of cell decompositions, the Steenrod axioms --homotopy functor, exactness, excision (based on transversality), and dimension-- and the geometric picture of a cycle or closed object representing a homology class, e.g. a closed oriented manifold with singularities in codimension at least two.
In the spring semester (MAT 541) there will be an independent treatment of cohomology as part of Homotopy Theory and Algebra, primarily as obstructions to building certain objects by induction: maps into target spaces, homotopies between maps, cross sections to projections of fibrations. Characteristc classes and k-invariants are direct applications of these examples.
Similarly cohomology classes arise in making inductive constructions in an algebraic setting: deforming the product in a commutative, Lie, or associative algebra; constructing the BRST differential of gauge theory in Physics; understanding the algebraic models of rational homotopy theory based on differential forms. These latter models are free graded commutative algebras with differential and they can be viewed as strong homotopy versions of Lie algebras called Lie-infinity algebras. Surprisingly, it will be clearer and easier to discuss the deformations and their obstructions in the more natural setting of commutative infinity, Lie infinity, and associative infinity algebras.
If time permits we will construct, as an application to Analysis, an infinity algebra structure on distributions and currents using smoothing and these ideas.
January 11, 2006