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 $H_q:=H_q(N;\mathbb{Z})$. The main result is a complete readily calculable classification of embeddings $N\to\mathbb{R}^7$, up to equivalence generated by isotopies and embedded connected sums with embeddings $S^4\to\mathbb{R}^7$. Such a classification was earlier known only for $H_1=0$ by Boéchat–Haefliger–Hudson 1970. Our classification involves the Boéchat–Haefliger invariant $\varkappa(f)\in H_2$, Seifert bilinear form $\lambda(f)\colon H_3\times H_3\to\mathbb{Z}$ and $\beta$-invariant assuming values in the quotient of $H_1$ defined by values of $\varkappa(f)$ and $\lambda(f)$. In particular, for $N=S^1\times S^3$ we define geometrically a $1$–$1$ correspondence between the set of equivalence classes of embeddings and an explicitly defined quotient of $\mathbb{Z}\oplus\mathbb{Z}$.
Our proof is based on development of Kreck modified surgery approach, involving some elementary reformulations, and also uses parametric connected sum.