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). The main result is a complete readily calculable classification of embeddings N -> R-7, up to equivalence generated by isotopies and embedded connected sums with embeddings S-4 -> R-7. Such a classification was earlier known only for H-1 = 0 by Boechat-Haeiger-Hudson 1970. Our classification involves the Boechat-Haeiger invariant x(f) is an element of H-2, Seifert bilinear form lambda(f) : H-3 x H-3 -> Z and beta-invariant assuming values in the quotient of H-1 defined by values of x(f) and lambda(f). In particular, for N = S-1 x S-3 we define geometrically a 1-1 correspondence between the set of equivalence classes of embeddings and an explicitly defined quotient of Z circle plus Z. Our proof is based on development of Kreck modified surgery approach, involving some elementary reformulations, and also uses parametric connected sum.