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).
Course description >
-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
-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
Operations with vector bundles
Classifying space of a vector bundle
Cohomology of the infinite Grassmanians
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)
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 >
- 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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