# A classification of smooth embeddings of 4-manifolds in 7-space, II - Mathematics > Geometric Topology

A classification of smooth embeddings of 4-manifolds in 7-space, II - Mathematics > Geometric Topology - Download this document for free, or read online. Document in PDF available to download.

Abstract: Let N be a closed, connected, smooth 4-manifold with H 1N;Z=0. Our mainresult is the following classification of the set E^7N of smooth embeddingsN->R^7 up to smooth isotopy. Haefliger proved that the set E^7S^4 with theconnected sum operation is a group isomorphic to Z {12}. This group acts onE^7N by embedded connected sum. Boechat and Haefliger constructed aninvariant BH:E^7N->H 2N;Z which is injective on the orbit space of thisaction; they also described imBH. We determine the orbits of the action: foru in imBH the number of elements in BH^{-1}u is GCDu-2,12 if u isdivisible by 2, or is GCDu,3 if u is not divisible by 2. The proof is basedon a new approach using modified surgery as developed by Kreck.

Author: **Diarmuid Crowley, Arkadiy Skopenkov**

Source: https://arxiv.org/