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 only for $H_1=0$ by the work of Bo'echat, Haefliger and Hudson from 1970. Our classification involves the Bo'echat-Haefliger invariant $\varkappa(f)\in H_2$,Seifertbilinearform$\lambda(f):H_3\times H_3\to Z$and$\beta$-invariant$\beta(f)$whichassumesvaluesinaquotientof$H_1$definedbyvaluesof$\varkappa(f)$and$\lambda(f)$\mathrm{.}Inparticular,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 development of Kreck modified surgery approach, involving some simpler reformulations, and also uses parametric connected sum.