Classification of Smooth Embeddings of 3-manifolds in the 6-space

We work in the smooth category. If there are knotted embeddings S → R, which often happens for 2m < 3n+4, then no concrete complete description of embeddings of n-manifolds into R up to isotopy was known, except for disjoint unions of spheres. Let N be a closed connected orientable 3-manifold. Our main result is the following description of the set Emb(N) of embeddingsN → R6 up to isotopy. The Whitney invariant W : Emb(N) → H1(N ;Z) is surjective. For each u ∈ H1(N ;Z) the Kreck invariant ηu : W−1u → Zd(u) is bijective, where d(u) is the divisibility of the projection of u to the free part of H1(N ;Z). The group Emb(S3) is isomorphic to Z (Haefliger). This group acts on Emb(N) by embedded connected sum. It was proved that the orbit space of this action maps under W bijectively to H1(N ;Z) (by Vrabec and Haefliger’s smoothing theory). The new part of our classification result is determination of the orbits of the action. E. g. for N = RP 3 the action is free, while for N = S1 × S2 we construct explicitly an embedding f : N → R6 such that for each knot l : S3 → R6 the embedding f#l is isotopic to f . The proof uses new approaches involving the Kreck modified surgery theory or the Boéchat-Haefliger formula for smoothing obstruction.