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The course will be devoted to the topics around the Bott periodicity theorem, which plays a key role in $K$-theory. The main attention will be paid to the constructions of Thom classes of vector bundles in $K$- and $KO$-theories and the general Riemann-Roch theorem in the Grothendieck form, which allows to extract integrality theorems for Hirzebruch genera from these constructions.
On the one hand, the course will be quite advanced: I will assume that the students are familiar with the theory of ordinary cohomology, including basic knowledge of characteristic classes and cohomology operations (although I will try to formulate all theorems I need carefully).
On the other hand, the course will be very basic and classical: it is planned to talk about the fundamental results of the 1950s and 1960s, which play a fundamental role in modern algebraic topology. The topic of the Riemann-Roch theorem is on the border between algebraic geometry and algebraic topology; in this course we will look at this theorem from the topological side.
Program.
- Complex Bott periodicity theorem: various formulations.
- Spectra, extraordinary (co)homology theories. $K$-theory as a cohomology theory.
- Sketch of the original proof of the Bott periodicity theorem through Morse theory.
- Atiyah's proof of the Bott periodicity theorem via approximating the clutching functions for vector bundles on $2$-sphere by trigonometric polynomials.
- Orientation of a vector bundle with respect to a cohomology theory. Thom class and Thom isomorphism. $K$-orientations of complex vector bundles.
- Chern classes. Splitting principle. Chern character.
- Riemann-Roch-Hirzebruch-Grothendieck theorem. Hirzebruch genera. Todd genus and its integerness.
- Algebraic bases of the real Bott periodicity theorem: Clifford algebras, spinor groups, their representations.
- Real Bott periodicity theorem, $KO$-theory.
- $K$-orientation of $\mathrm{Spin}^{\mathbb{C}}$-bundles and $KO$-orientation of spinor bundles.
- Integrality theorems. Rohlin's theorem on the signature of a spinor four-dimensional manifold.
- Hattori-Stong theorem.
 RSS: Forthcoming seminars
 Lecturer
Gaifullin Alexander Aleksandrovich
Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center |
