  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 $\varkappa(f)\in
H_2$, Seifert bilinear form $\lambda(f):H_3\times H_3\to Z$ and
$\beta$-invariant $\beta(f)$ which assumes values in a quotient of $H_1$
defined by values of $\varkappa(f)$ and $\lambda(f)$. In particular, 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.
