A classification of smooth embeddings of 4-manifolds in 7-space, II
Abstract
Let N be a closed, connected, smooth 4-manifold with H_1(N;Z)=0. Our main result is the following classification of the set E^7(N) of smooth embeddings N->R^7 up to smooth isotopy. Haefliger proved that the set E^7(S^4) with the connected sum operation is a group isomorphic to Z_{12}. This group acts on E^7(N) by embedded connected sum. Boechat and Haefliger constructed an invariant BH:E^7(N)->H_2(N;Z) which is injective on the orbit space of this action; they also described im(BH). We determine the orbits of the action: for u in im(BH) the number of elements in BH^{-1}(u) is GCD(u/2,12) if u is divisible by 2, or is GCD(u,3) if u is not divisible by 2. The proof is based on a new approach using modified surgery as developed by Kreck.
Keywords
Cite
@article{arxiv.0808.1795,
title = {A classification of smooth embeddings of 4-manifolds in 7-space, II},
author = {Diarmuid Crowley and Arkadiy Skopenkov},
journal= {arXiv preprint arXiv:0808.1795},
year = {2018}
}
Comments
25 pages. Typos corrected and minor changes to improve exposition. To appear in the International Journal of Mathematics