Abstract:
In his famous 1967 paper, S. P. Novikov was the first to notice a deep connection between complex cobordism and formal group laws theory (subsequently fruitfully studied in the works of Soviet and foreign topologists). For example, Novikov's results imply that any complex genus (a homomorphism from the complex cobordism ring) with values in a ring $R$ is in one-to-one correspondence with a formal group law over $R$, and if $R$ is a $\mathbb Q$-algebra, then the exponential of the corresponding formal group law is a power series corresponding to the complex genus in the sense of Hirzebruch. I.M.Krichever (following Atiyah and Hirzebruch) introduced the notion of rigidity of a complex genus on a manifold with a torus action and defined the exponential of the universal elliptic genus, which is rigid on all SU-manifolds (with trivial first Chern class). V.M.Buchstaber, T.E.Panov and N.Ray interpreted the notion of rigidity in terms of cobordisms, which allowed to prove a localization formula in cobordisms (generalizing the Atiyah–Bott and Krichever results) in cobordisms that expresses the value of the equivariant extension of a genus on a manifold with a torus action in terms of fixed points of the action. This formula yields a rigidity equation that the exponential must satisfy for the genus to be rigid on a particular manifold with a torus action. I will discuss how this yields a characterization of the universal complex elliptic genus in terms of its rigidity on just two SU manifolds.
