Lecturer: Bernardo Uribe
The idea of bordism could be traced back to Poincaré. His idea was to develop a notion on which one studies manifolds via the manifolds that could be mapped into it, modulo the equivalence relation of being cobordant (these are called the bordism groups of a manifold). This notion was changed by the simplicial description, which turns out to be easier to calculate, and simplicial homology was born.
Thom and Pontrjiagin were the first to notice that these bordism groups could be calculated using stable homotopy. They proved that the bordism groups of a point could be calculated as the stable homotopy groups of certain limit of spaces. This marvelous result was the key element that permitted Milnor, Thom and others to calculate explicitly the bordism groups of a point, and therefore to determine the complete invariants of unoriented bordism and unitary bordism.
The equivariant version of bordism, when we consider a compact Lie group acting on a space and we consider equivariant mappings modulo the cobordism relation, turned out very difficult ot calculate and are still not known in many cases.
In these lectures I will introduce bordism, its propoerties and the main results in bordism that permit to calculate the bordism groups. Then I will focus on the equivariant version and I will explain what is known and what still is not known.
References:
P.E. Conner and E.E. Floyd, The relations of Cobordism to K-theories , Lecture notes in Mathematics 28, Springer Verlag, 1966
Stong, R. E. Complex and oriented equivariant bordism. 1970 Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) pp. 291–316 Markham, Chicago, Ill. (Reviewer: R. Schultz) 57.47
Freed, D. Bordism: OLD AND NEW: https://www.ma.utexas.edu/users/dafr/bordism.pdf