Embeddings of non-simply-connected 4-manifolds in 7-space. I. classification modulo knots

Abstract

We work in the smooth category. Let N be a closed connected orientable 4-manifold with torsion free H1, where Hq:=Hq(N; Z). The main result is a complete readily calculable classification of embed-dings N ? R7, up to equivalence generated by isotopies and embedded connected sums with embeddings S4 ? R7 . Such a classification was earlier known only for H1 = 0 by Boéchat–Haefliger–Hudson 1970. Our classification involves the Boéchat–Haefliger invariant ?(f) ? H2, Seifert bilinear form ?(f): H3 × H3 ? Z and ß-invariant assuming values in the quotient of H1 defined by values of ?(f) and ?(f). In particular, for N = S1 × S3 we define geometrically a 1–1 correspondence between the set of equivale..