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

Abstract

We work in the smooth category. If there are knotted embeddings S^n\to R^m, which often happens for 2m<3n+4, then no concrete complete description of embeddings of n-manifolds into R^m 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^6(N) of embeddings N\to R^6 up to isotopy. The Whitney invariant W : Emb^6(N) \to H_1(N;Z) is surjective. For each u \in H_1(N;Z) the Kreck invariant η_u : W^{-1}u \to Z_{d(u)} is bijective, where d(u) is the divisibility of the projection of u to the free part of H_1(N;Z). The group Emb^6(S^3) is isomorphic to Z (Haefliger). This group acts on Emb^6(N) by embedded connected sum. It was proved that the orbit space of this action maps under W bijectively to H_1(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=S^1\times S^2 we construct explicitly an embedding f : N \to R^6 such that for each knot l:S^3\to R^6 the embedding f#l is isotopic to f. Our proof uses new approaches involving the Kreck modified surgery theory or the Boechat-Haefliger formula for smoothing obstruction.

32 pages, a link to http://www.springerlink.com added, to appear in Math. Zeit

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• Classification of knotted tori
• Embeddings of non-simply-connected 4-manifolds in 7-space. I. Classification modulo knots
• On the Browder-Levine-Novikov embedding theorems
• Involutions on S^6 with 3-dimensional fixed point set
• Embeddings of k-connected n-manifolds into R^{2n-k-1}
• Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification
• An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number
• Classification of knotted tori in the 2-metastable dimension
• How do autodiffeomorphisms act on embeddings
• The classification of linked $3$-manifolds in $6$-space
• A new invariant and parametric connected sum of embeddings