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 \eta_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.Comment: 32 pages, a link to http://www.springerlink.com added, to appear in Math. Zei

The Classification of Certain Linked 3-Manifolds in 6-Space

“…All the homology groups in the text are with coefficients in Z unless another group is explicitly specified. For any closed connected orientable 3-manifold N the Whitney invariant W : E 6 (N ) → H 1 (N ) is defined in [Sk08a]. We give an equivalent definition in §1.6.…”

Классификация Вложений Торов В 2-Метастабильной Размерности

“…[18]- [20]), известно о вложениях в размерности ниже метастабильной немного: все известные явные классификационные результаты касаются узлов и зацеплений (они перечисле-ны выше), заузленных торов в размерности m = p + 3 2 q + 3 2 , 3-мерных многооб-разий в R 6 и 4-мерных многообразий в R 7 (см. соответственно [21], [22] и [9]). Сформулируем основной "практический" результат работы, анонсированный в [11], -явный критерий конечности множества изотопических классов заузлен-ных торов в 2-метастабильной размерности (см.…”

Embeddings of $k$-connected $n$-manifolds into $\mathbb {R}^{2n-k-1}$

“…∂(N × I)) to an isotopy. Construction 2.6 of ψ (analogous to [Sk081], proof of the surjectivity of W in §5).…”