Topology of Manifolds
Alexander Mishchenko and
Definitions and basic results of cobordism theory can be found in the survey articles by T.Panov
in "The Manifold Atlas Project":
The achievements of the Novikov school laid the foundations of modern theory of cobordism. Key results on the structure of cobordism rings were obtained,
and the formal group law techniques in cobordism theory were developed in the 1960s and 1970s.
Nowadays cobordism theory remains one of the main subjects of research at our department. Studies have focused on the following main areas:
These methods rely upon the Atiyah-Singer index theorem and the generalised Lefschetz fixed point formula, and make use of the formal group law techniques developed
by the Novikov school in the 1960-70s.
Application of cobordism and formal group laws to the study of finite group actions on manifolds
- Study of topological invariants of manifolds (in particular, characteristic classes and Hirzebruch genera) by the methods of cobordism and index theory
Furthermore, cobordism of manifolds with group action play important role in Toric Topology. Buchstaber, Panov and Ray proved that each complex cobordism class contains a quasitoric manifold, and therefore the quasitoric analogue of the Hirzebruch problem (on connected algebraic representatives in cobordism classes) has positive solution.