The idea of this two semester course is to select from all of algebraic topology those parts that are quite useful in geometry,,algebra and analysis.

Homology theory is presented in terms of geometric cycles and homologies mapping into a space. Two related topics are then spherical cycles relating to the Hurewicz theorems and the interplay of the intersection product in manifolds with the dualities of Alexander, Lefschetz, and Poincaré.

Cohomology appears naturally as obstructions in inductive constructions of mappings and cross sections of bundles. Characteristic classes are an immediate corollary as are Postnikov systems. Cohomology also appears naturally as obstructions to the term by term construction of deformations of algebraic structures such as (differential, graded) associative algebras, commutative algebras or lie algebras. These obstruction theories only obtain full expression when applied to the derived version of the deformation problem where exact equations like associativity or Jacobi or commutativity are replaced by Stasheff type hierarchies of chain homotopies correcting the inequations.

The latter ideas have appeared at the deeper aspects of the interface of geometry and algebra in differential topology, symplectic topology, and holomorphic topology. (mirror conjectures)

Differential forms relate analysis to cohomology and rational homotopy. The above algebraic ideas are useful for extending the partial product on currents to a meaningful structure.

The course will emphasize the fundamental aspects of these issues. Grades will be based on (graded by me) homework assignments, class participation and the exams.