SoSe 21 Seminar: Topological K-theory
Time and place: Monday 12-14 online
Topological K-theory was historically the first example of a generalized cohomology theory for spaces. This means that it satisfies all the Eilenberg–Steenrod axioms for homology, except that the K-theory of a point is not concentrated in degree 0. After Grothendieck introduced the K-group K₀ of algebraic varieties in 1957, Atiyah and Hirzebruch quickly realized that an analogous definition in topology was interesting. For X a topological space, they defined K⁰(X) to be the group completion of the monoid of isomorphism classes of complex vector bundles over X. Coincidentally, in 1957, Bott had just discovered a surprising periodicity phenomenon in the homotopy groups of the orthogonal and unitary groups, which is now known as Bott periodicity. This allowed Atiyah and Hirzebruch to extend K⁰(X) to a full-fledged generalized cohomology theory K*(X).
The first half of this seminar is dedicated to the definition of topological complex K-theory as a generalized cohomology theory. We will study vector bundles on topological spaces and prove Bott’s periodicity theorem. The second half will cover some applications and miscellaneous topics. We will construct the Adams operations and prove that real division algebras only exist in dimensions 1,2,4,8. We will also introduce the Chern character and the connection with Fredholm operators and index theory.
|Vector bundles I: definitions and constructions
|Vector bundles II: paracompact spaces
|Vector bundles III: the classification theorem
|The Grothendieck ring of vector bundles
|The splitting principle
|Ritheesh Krishna Thiruppathi
|Adams operations and the Hopf invariant one theorem
|The Chern character
|The Atiyah–Jänich theorem