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 complete readily calculable classification of embeddings $N\to R^7$, up to equivalence relation generated by isotopy and embedded sum $S^4\to R^7$. Such was already known only for $H_1=0$ Boechat, Haefliger Hudson from 1970. involves Boechat-Haefliger invariant $\varkappa(f)\in H_2$, Seifert bilinear form $\lambda(f):H_3\times H_3\to Z$ $\beta$-invariant $\beta(f)$ which assumes values quotient $H_1$ defined $\varkappa(f)$ $\lambda(f)$. In particular, $N=S^1\times S^3$ we give geometrically 1-1 correspondence between set classes an explicit $Z\oplus Z$. Our proof based on development Kreck modified surgery approach, involving some simpler reformulations, also uses parametric sum.