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.

1.   Complex Bott periodicity theorem: various formulations.
2. Spectra, extraordinary (co)homology theories. $K$-theory as a cohomology theory.
3. Sketch of the original proof of the Bott periodicity theorem through Morse theory.
4. Atiyah's proof of the Bott periodicity theorem via approximating the clutching functions for vector bundles on $2$-sphere by trigonometric polynomials.
5. Orientation of a vector bundle with respect to a cohomology theory. Thom class and Thom isomorphism. $K$-orientations of complex vector bundles.
6. Chern classes. Splitting principle. Chern character.
7. Riemann-Roch-Hirzebruch-Grothendieck theorem. Hirzebruch genera. Todd genus and its integerness.
8. Algebraic bases of the real Bott periodicity theorem: Clifford algebras, spinor groups, their representations.
9. Real Bott periodicity theorem, $KO$-theory.
10. $K$-orientation of $\mathrm{Spin}^{\mathbb{C}}$-bundles and $KO$-orientation of spinor bundles.
11. Integrality theorems. Rohlin's theorem on the signature of a spinor four-dimensional manifold.
12. Hattori-Stong theorem.

Lecturer
Gaifullin Alexander Aleksandrovich

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, agreement no. 075-15-2022-265).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center