A classification of smooth embeddings of 3-manifolds in 6-space
| dc.creator | Skopenkov, A. | |
| dc.date | 2006-03-17 | |
| dc.date | 2008-01-15 | |
| dc.date.accessioned | 2026-07-07T12:09:31Z | |
| dc.date.available | 2026-07-07T12:09:31Z | |
| dc.description | 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. | |
| dc.description | 32 pages, a link to http://www.springerlink.com added, to appear in Math. Zeit | |
| dc.identifier | https://arxiv.org/abs/math/0603429 | |
| dc.identifier | http://arxiv.org/abs/math/0603429 | |
| dc.identifier | Math. Zeitschrift, 260:3 (2008) 647-672 | |
| dc.identifier | doi:10.1007/s00209-007-0294-1 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/209653 | |
| dc.subject | Geometric Topology | |
| dc.subject | Algebraic Topology | |
| dc.subject | 57R40; 57R52 | |
| dc.title | A classification of smooth embeddings of 3-manifolds in 6-space | |
| dc.type | text |