Algebraic Topology II: Topological K-Theory (Spring 2015)

This course is taught jointly by Javier J. Gutiérrez, Ieke Moerdijk, and Matan Prasma.
The teaching assistant for this course is Joost Nuiten.

  • Time and place: Thursdays 10:30-13:30, Radboud Universiteit Nijmegen, room HG03.085 (Huygensgebouw).
  • Prerequisites: The master course in Algebraic Topology.

Course description >

Topological K-theory, the first generalized cohomology theory to be studied thoroughly, was introduced in a 1961 paper by Atiyah and Hirzebruch, where they adapted the work of Grothendieck on algebraic varieties to a topological setting. Since that time, topological K-theory has become a powerful and indispensable tool in topology, differential geometry, and index theory.

We will begin by studying the theory of vector bundles and related algebraic notions, followed by the definition of K-theory and proofs of some important theorems in the subject, such as the Bott periodicity theorem. Some of the topics that might be covered in the course include:

  • Basics about vector bundles. Definition of K(X), K-n(X).
  • Homotopy invariance and long exact sequence
  • Eilenberg-Steenrod axioms
  • Operations with vector bundles
  • Classifying space of a vector bundle
  • Cohomology of the infinite Grassmanians
  • Bott periodicity
  • Clifford algebras
  • Adams operations and lambda-rings

Lecture notes >

  • Lecture 6: K-theory as a cohomology theory. (12/03)
  • Lecture 7: K-theory groups of the spheres. (19/03)
  • Lecture 8: The complex K-theory spectrum. (26/03)
  • Paper by B. Harris, Bott periodicity via simplicial spaces, J. Algebra, 62 (1980), 450–454.
  • Paper by I. Moerdijk, Bisimplicial sets and the group-completion theorem, Algebraic K-theory: connections with geometry and topology (Lake Louise, AB, 1987), 225–240, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 279, 1989.

Exercise sheets >

References >

  • M. F. Atiyah, K-theory. Lecture notes by D. W. Anderson W. A. Benjamin, Inc., New York-Amsterdam 1967
  • A. Hatcher, Vector bundles and K-theory, Book project available at the author's webpage.
  • D. Husemoller, Fibre bundles. Graduate Texts in Mathematics, 20. Springer-Verlag, New York, 1994.
  • M. Karoubi, K-theory. An introduction. Grundlehren der Mathematischen Wissenschaften, Band 226. Springer-Verlag, Berlin-New York, 1978.

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