Cobordism theories: An algebraic journey from topology to geometry

Abstract

Cobordism theories have been studied for a long time in various forms and guises. Many of these theories were shown to have a very nice structure and became important tools in the study of differentiable manifolds. When Quillen gave an axiomatic description for the theory of complex cobordism, it became possible to define an equivalent tool for the language of schemes. Levine and Moore translated Quillen's axioms into algebraic geometry and defined algebraic cobordism. This theory has strong relations with the Chow group and K-theory, just like cobordism theories in algebraic topology relate to homology and K-theory of vector bundles. Algebraic cobordism was given a geometric interpretation by Levine and Pandharipande. Lee and Pandharipande were able to extend this to a theory of schemes with bundles, similar to the theory Atiyah and Singer used in their proof of the Atiyah-Singer index theorem. Using this theory Tzeng was able to prove conjectures of Vainsencher and Göttsche about nodal curves on surfaces. This generalises the result of Fomin and Mikhalkin that the number of nodal curves through a specific number of points on the projective plane is a polynomial in the degree of the curve, to an arbitrary surface.