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

D. Crowley, A. Skopenkov

Published 2016-12-14Version 1

We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q:=H_q(N;Z)$. Our main result is a readily calculable classification of embeddings $N\to R^7$ 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 $S^4\to R^7$ acts on the set of embeddings $N\to R^7$ 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, the classification involves a refinement of the Kreck invariant from our work of 2008 which detects the action of knots. 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 quotient of the set $Z\oplus Z\oplus Z_{12}$.