Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification

TL;DR

This paper provides a comprehensive smooth classification of embeddings of certain 4-manifolds into 7-space, extending previous results to cases with nontrivial first homology and describing the action of knots on these embeddings.

Contribution

It introduces a complete classification method for embeddings of 4-manifolds with torsion-free first homology into R^7, including the description of knot actions for manifolds with nonzero H_1.

Findings

Classification is complete when H_2=0 or signature conditions are met.

Describes the action of knots on embeddings with an indeterminacy.

Provides a geometric correspondence for embeddings of S^1×S^3 into R^7.

Abstract

We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q := H_q(N; \mathbb Z)$. Our main result is a readily calculable classification of embeddings $N\to\mathbb R^7$ up to isotopy, with an indeterminancy. Such a classification was only known before for $H_1=0$ by our earlier work from 2008. Our classification is complete when $H_2=0$ or when the signature of $N$ is divisible neither by 64 nor by 9. The group of knots $S^4\to\mathbb R^7$ acts on the set of embeddings $N\to\mathbb R^7$ up to isotopy by embedded connected sum. In Part I we classified the quotient of this action. The main novelty of this paper is the description of this action for $H_1\ne0$, with an indeterminancy. Besides the invariants of Part I, detecting the action of knots involves a refinement of the Kreck invariant from our work of 2008. For $N=S^1\times S^3$ we give a geometrically defined 1--1 correspondence between the set of isotopy classes of embeddings and a certain explicitly defined quotient of the set $\mathbb Z\oplus\mathbb Z\oplus\mathbb Z_{12}$.