Obstruction theory for the existence of 4-dimensional $\xi$-fillings of 3-manifolds

By a result of Milnor we know that every spin 3-manifold spin bounds a 4-manifold. However, if we require this null-bordism to be of a prescribed normal 1-type, then it may no longer exist. In the spin case this normal $1$-type is a spin structure on the 3 manifold Y, together with the map on fundamental groups $\pi_1(Y) \to \pi_1(X)$ that we wish to realise via a bounding $4$-manifold $X$. We describe a three-stage geometric obstruction theory for the existence of a filling which extends this structure. Our main contribution is the definition of a new `tertiary' obstruction defined using Wall's equivariant self-intersection number. This is joint work with Daniel Galvin and Peter Teichner.