Embedding four manifolds in \(\mathbb{R}^ 7\) (Q1332839)

\textit{A. Haefliger} and \textit{M. W. Hirsch} proved that a closed smooth \(n\)-manifold \((n \neq 4)\) can be embedded smoothly into \(\mathbb{R}^{2n - 1}\) if and only if the normal Stiefel-Whitney class \(\overline w_{n - 1} (M)\) vanishes [Topology 2, 129-135 (1963; Zbl 0113.386)]. The author studies the problem of embedding closed 4-manifolds into \(\mathbb{R}^ 7\). His main results are the following. (A) A smooth closed 4-manifold embeds smoothly into \(\mathbb{R}^ 7\) if and only \(\overline w_ 3 (M) = 0\). (B) A topological closed 4-manifold embeds locally flat into \(\mathbb{R}^ 7\) if and only if (i) in the nonorientable case \(\overline w_ 3 (M) = 0\) and \(KS (M) = 0\), (ii) in the orientable case, there exists a class \(w \in H = H^ 2 (M; \mathbb{Z})/\)torsion so that \[ w^ 2 = \text{sign} (M),\;wx = x^ 2 \bmod 2 \quad \text{for every} \quad x \in H, \quad \text{and } KS (M) = 0, \] where \(KS (M) \in \mathbb{Z}_ 2\) is the Kirby-Siebenmann obstruction. Part (A) generalizes a result of \textit{J. Boéchat} and \textit{A. Haefliger} [in `Essays Topol. Related Topics, Mem. dédiés à Georges de Rham' 156- 166 (1970; Zbl 0199.270)] to the case of nonorientable manifolds.