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 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 K^0(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 K^0(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.

• Organisational meeting/distribution of topics: Wednesday February 10 at 16:15 on Zoom (meeting
  ID: 839 3217 6151, passcode: Atiyah). If you cannot make it to the meeting or have any
  questions, please contact me by email.
• Registration for course work/examination/ECTS: FlexNow

• Presentation: Giving a seminar talk of roughly 90 minutes

• Detailed written report of the seminar talk