Topology and Geometry Seminar

Title: Introduction to topological K-theory

Abstract: Topological K-theory is an invariant of spaces based on vector bundles. We will review vector bundles and their representability by Grassmannians. We will then introduce the group K^0(X), which measures to what extent vector bundles on X can be non-trivial. Finally, Bott periodicity will allow us to extend the group K^0(X) to a cohomology theory.

Title: Introduction to topological K-theory, part 2

Abstract: In this second part, we will further discuss Stiefel manifolds and Grassmannians, as well as stabilization of vector bundles. Next, we will introduce the group K^0(X) along with its reduced version, and compute some examples. Time permitting, we will use Bott periodicity to extend the group K^0(X) to a cohomology theory.

Title: Introduction to topological K-theory, part 3

Abstract: Now that we have the K-theory group K(X) at our disposal, our next goal will be to compute it for the circle S^1 and the sphere S^2. We will then discuss Bott periodicity and extend the group K(X) to a cohomology theory.

Title: The Chern character and its Applications

Abstract: In this talk we will first review some of the properties of Chern classes and Steifel Whitney classes. Next, we will introduce the Chern character, now that we have made K(X) into a cohomology theory using Bott periodicity, in the earlier talk. Following that, we will see some interesting applications of the Chern character map and hence compute the unreduced K-theory of CP^n.

Title: The Chern character: Epilogue and Prelude

Abstract: At this point we have two cohomology theories in hand – HZ (singular cohomology) and KU (complex K-theory) – and the Chern character ch establishes a comparison between them. Even the first properties of ch can already give rise to interesting consequences, as we saw last time. 

The story, however, does not simply stop here. Both HZ and KU are “very geometric”, as they support Thom classes for complex vector bundles. This talk will invite us to take a further (geometric) look at the comparison ch: how does it compare the K-theoretic Thom class with the ordinary cohomological one? Exploring this question will lead us to define the Todd class as the correction term, and to introduce a topological version of the Grothendieck-Riemann-Roch theorem.

Additional advertisement for topologists: This talk will open the doors to a number of important future topics, including the e-invariant and the J-homomorphism.

Additional advertisement for number theorists: Bernoulli numbers will appear (multiple times) in the talk.

Title: Adams operations

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Title: The Serre-Swan theorem

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