Embeddings of non-simply-connected 4-manifolds in 7-space. I. Classification modulo knots

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 complete readily calculable classification of embeddings $N\to R^7$, up to the equivalence relation generated by isotopy and embedded connected sum with embeddings $S^4\to R^7$. Such a classification was already known when $H_1=0$ by the work of Bo\'echat, Haefliger and Hudson from 1970. Our results for $H_1\ne0$ are new. The classification involves the Bo\'echat-Haefliger invariant $\varkappa(f)\in H_2$, and two new invariants: a Seifert bilinear form $\lambda(f):H_3\times H_3\to Z$ and $\beta$-invariant $\beta(f)$ which assumes values in a quotient of $H_1$ depending on the values of $\varkappa(f)$ and $\lambda(f)$. For $N=S^1\times S^3$ we give a geometrically defined 1-1 correspondence between the set of equivalence classes of embeddings and an explicit quotient of the set $Z\oplus Z$. Our proof is based on Kreck's modified surgery approach to the classification of embeddings, and also uses parametric connected sum.