Topology Seminar

It is a classical result, due to Milnor, that every spin 3-manifold spin bounds a 4-manifold.  If we ask for this null-bordism to be of a prescribed normal 1-type, i.e. its stable normal bundle has a prescribed Postnikov-Moore approximation, then it may no longer exist.  We fix this data on a given 3-manifold and describe a three stage geometric obstruction theory for the existence of a filling which extends this structure, which we show coincide with the obstructions provided by the James spectral sequence.  If we assume the first two obstructions vanish, we can reduce the problem to a certain variation on a sphere embedding problem in a 4-manifold.  Our main contribution is a new `tertiary' obstruction, defined using Wall's equivariant self intersection number, that entirely governs this embedding problem.  This is joint work with Peter Teichner and Simona Veselá.