As part of the Simons Center program, Interactions of Homotopy Theory and Algebraic Topology with Physics through Algebra and Geometry, John Morgan will give a series of lectures (8 to 10 lectures) on Sheaf Theory with applications to duality. The course will be aimed at intermediate graduate students and above. The only prerequisite is a basic course in algebraic topology.
These lectures will be on Fridays at 2:45pm, in the Simons Center seminar room, 313 beginning Friday October 3.
Title: A Topologist looks at Sheaf Theory
Sheaf theory has long been an essential tool in algebraic geometry, algebraic number theory, and complex analysis, but its inspiration comes directly from topology. This lecture course will emphasize these roots, hopefully making sheaf theory seem natural to those with a topological bent. The course will begin by covering the basic topics in sheaf theory describing the objects and the four basic maps of the theory and then will culminate with a discussion of Verdier duality, which generalizes Poincare duality.
This theory will then be applied to define a bordism theory, called duality bordism, whose coefficient group agrees with the Grothendieck group of chain complexes satisfying Poincare duality modulo those that sit as the boundary term in an exact sequence satisfying Lefschetz duality. This bordism group is the Pontryjagin dual homology theory to the cohomology theory associated with surgery theory. This means that a surgery problem is completely classified by evaluating surgery obstructions (signatures, and Arf invariants) of its restrictions to all possible duality bordism elements.
Direct analysis of this bordism theory allows one to identify it at odd primes with real K-theory and at the prime 2 with ordinary homology.