Cobordism invariants and characteristic classes

Abstract

The concept of cobordism owes its origin to the Frenchman Henri Poincar e. The relation of cobordism is de ned for two closed manifolds M and N with same dimensión n. By de nition M And Nare cobordant if there is a manifold of di mension n+ 1 whose boundary is MtN; that is, the disjoint union of M and N. Being a relation of equivalence weaker than homeomorphism and di eomor-phism, the search for a classi cation of manifolds under this one seems easier to describe. This concept has a leading role in geometric topology and algebraic topology thanks to contributions from mathematicians such as Ren e Thom and Lev Pontrjagin. More recently, it has been a central part of the development of topological theory of quantum elds. Thom in the mid-twentieth century proved the theorem exhibiting an isomorphism between the ring of equivalence classes of cobordism and the homotopy group of the Thom spectrum, being this one of his most important contributions in geometry. This theorem allows us, for example, to describe the torsion free part of the ring of cobordism as a polynomial ring in in nite many variables, a really impressive result