A classification of smooth embeddings of 4-manifolds in 7-space, II

TL;DR

This paper classifies smooth embeddings of closed, connected 4-manifolds with trivial first homology into 7-space, extending previous work by analyzing the action of a Z_{12} group and using modified surgery techniques.

Contribution

It provides a detailed classification of embeddings of 4-manifolds into 7-space, determining orbit counts under a group action using a new modified surgery approach.

Findings

The set of embeddings forms a structured classification with explicit orbit counts.

The action of the Z_{12} group on embeddings is fully characterized.

A new proof technique using Kreck's modified surgery is developed.

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.