Pirates of the Cobordism: Curse of the Knot Surgery

Two n-dimensional manifolds M1 and M2 are cobordant if there is an (n + 1)-dimensional manifold W with the disjoint union of M1 and M2 as its boundary. Such a simple definition can lead to very interesting results in many different areas of mathematics. Algebraic topology is perhaps the most obvious one, with cobordism being central to surgery theory and the study of high-dimensional manifolds. Cobordisms and Morse Theory have a very intimate relationship, whilst at the same time cobordisms are having fun being the domains of TQFTs (topological quantum field theories). Cobordism theories can be made into extraordinary cohomology theories, and they are also playing a central role in modern-day knot theory. Contrary to what Wikipedia says, cobordism has nothing to do with “a Danish modern art movement based on gluing carboard boxes to your face”. In this talk I will show why 1,2 and 3-manifolds are all null-cobordant (i.e. cobordant to an empty manifold), with the 3-dimensional case obviously taking the majority of the time. Interestingly, the proof hinges on showing that we can construct every 3-manifold from S3 by surgery on a link, a result which is very beautiful in its own right.