Recent progress in cobordism theory

Cobordism is a basic tool for the classification of manifolds (i.e. topological spaces locally homeomorphic to a Euclidean space). Early success in calculating several variants of cobordism by Thom, Milnor, Novikov and others in the 1950's-1960's became one of the starting points of modern algebraic topology. Among one of the most well-known cases still not solved is symplectic cobordism. I will discuss a precursor to that theory, called self-conjugate cobordism, which was recently solved in a joint paper by myself, Po Hu, Petr Somberg, and Benjamin Riley. The solution is in the form of the homology of a particular chain complex, which, while explicitly algorithmic, is very difficult to carry out computationally. I will briefly describe this complex, with an emphasis on highlighting some gaping omissions in existing computer algebra systems, addressing which would greatly advance practical computations such as the present one.