A classification of smooth embeddings of 3-manifolds in 6-space

We work in the smooth category. If there are knotted embeddings $${S^n \to \mathbb{R}^m}$$, which often happens for 2m < 3n + 4, then no explicit complete description of the embeddings of n-manifolds into $${\mathbb{R}^m}$$ up to isotopy was known, except for the disjoint unions of spheres. Let N be a closed connected orientable 3-manifold. Our main result is the following description of the set Emb6(N) of embeddings $${N \to \mathbb{R}^6}$$ up to isotopy. We define the Whitney and the Kreck invariants and prove that the Whitney invariant $${W: {\rm Emb}^6(N) \to H_1(N; \mathbb{Z})}$$ is surjective. For each$${u \in H_1(N; \mathbb{Z})}$$ the Kreck invariant $${\eta_u : W^{-1} u \to \mathbb{Z}_{d(u)}}$$ is bijective, where d(u) is the divisibility of the projection of u to the free part of$${H_1(N; \mathbb{Z})}$$. The group Emb6(S3) is isomorphic to $${\mathbb{Z}}$$ (Haefliger). This group acts on Emb6(N) by embedded connected sum. It was proved that the orbit space of this action maps under W bijectively to $${H_1(N; \mathbb{Z})}$$ (by Vrabec and Haefliger’s smoothing theory). The new part of our classification result is the determination of the orbits of the action. E.g. for $${N = \mathbb{R}P^3}$$ the action is free, while for N = S1 × S2 we explicitly construct an embedding $${f : N \to \mathbb{R}^6}$$such that for each knot$${l : S^3 \to \mathbb{R}^6}$$the embedding f#l is isotopic to f. The proof uses new approaches involving modified surgery theory as developed by Kreck or the Boéchat–Haefliger formula for the smoothing obstruction.