Orientable 4-manifolds topologically embed into R7

In this paper, we prove that a 4-manifold topologically embeds into R7 if and only if its third normal Stiefel–Whitney class vanishes. In particular, this shows that orientable 4-manifolds topologically embed into R7 i.e. the hard Whitney embedding theorem holds in dimension 4 in the topological category. The former generalizes the classical Haefliger–Hirsch theorem to dimension 4 in the topological category (Compare Fang, Topology 33 (1994) 447), and the latter answers an open problem in Kirbyʹs list (Kirby, in W.H. Kozez (Ed.), Geometric Topology, Vol. 2, International Press, 1997, 35, Problem 4.19). Comparing with our previous work (Fang, Topology 33 (1994) 447) we obtain infinitely many simply connected closed 4-manifolds that can be locally flatly embedded in R7 but can not have normal bundles.