Embeddings of 4-manifolds into 7-space

Abstract: (joint with Arkadiy Skopenkov). Let $N$ be a closed connected smooth n-manifold and let $E^m(N)$ be the set of isotopy classes of embeddings of $N$ into Euclidean m-space. The set $E^{n+2}(S^n)$ of isotopy classes of codimension-2 embeddings of the n-sphere has been intensively studied. In the 60s and 70s a great deal was also learnt about embeddings of closed manifolds in codimension-3 and higher: key names are Haefliger and Wall amongst others. The results of the 60s and 70s were more complete in the piecewise linear (PL) category and tended to be general results as opposed to detailed analyses of specific examples. Recently there has been important new progress based on a modified surgery algorithm for classifying embeddings discovered by M. Kreck. In this talk I take up the problem of determining $E^7(N)$ for a closed smooth connected orientable 4-manifold N. In the case that the integral homology of groups of N are torsion free I will present a complete description of $E^7(N)$ via readily computable invariants. I will also present a complete description of the set of the PL embeddings of N up to PL isotopy. The case $N = S^1 \times S^3$ is of particular interest: it gives a counter example to the ``informal $\alpha$-invariant hypothesis" as well as exhibiting subtle inertial phenomena.