Integrable embeddings and foliations (Q2892855)

Let \(M\) be an open smooth \(n\)-manifold, and \(L\) a properly embedded \(k\)-submanifold of \(M\) with codimension \(m=n-k\), whose normal bundle is denoted by \(\nu(L)\). It is said that \(L\) is (1)~weakly integrable (WI) if there is a submersion to the \(m\)-Euclidean space, \(\Phi:M\to\mathbb{E}^m\), such that \(h(L)\subset\Phi^{-1}(0)\); (2)~strongly integrable (SI) when we can get \(M=\Phi^{-1}(0)\) as above; (3)~a complete intersection (CI) if there is a smooth map \(\Xi:M\to\mathbb{E}^m\) such that \(0\) is a regular value and \(\Xi^{-1}(0)=M\); and (4)~totally weakly integrable (TWI) if, for any \(n\geq k+2\), any embedding \(L\to\mathbb{E}^n\) is WI if and only if \(\nu(L)\) is trivial.NEWLINENEWLINEThe paper is mainly a systematic and thorough study of these concepts, obtaining nice characterizations of the manifolds \(L\) that satisfy these conditions. As particular cases, this implies some previous results of \textit{G. Hector} and \textit{W. Bouma} [Indag. Math. 45, 443--452 (1983; Zbl 0529.57012)], \textit{N. Watanabe} [Topology 32, No.2, 251-257 (1993; Zbl 0784.57004)] and \textit{S. Miyoshi} [ibid. 34, No. 2, 383--387 (1995; Zbl 0845.57024)].NEWLINENEWLINEWhen \(m=1\), let \(\hat M_L\) be the manifold obtained by cutting \(M\) along \(L\), and let \(H^\infty_*(M)\) be the homology of the complex of locally finite infinite simplicial chains in \(M\). It is shown that the following conditions are equivalent: (1)~\(L\) is SI; (2)~\(L\) is \(WI\) and \(0=[L^\epsilon]\in H^\infty_{n-1}(M)\) for some orientation \(\epsilon\) of \(L\); (3)~\(L\) is CI and WI; and (4)~\(L\) is CI and \(\hat M_L\) is open. Therefore the following conditions are equivalent if \(\beta_1(M)=0\): (1)~\(L\) is open; (2)~\(L\) is WI; and (3)~\(L\) is SI.NEWLINENEWLINEFor \(m\geq2\), the main tool is a relative version of the Philips-Gromov \(h\)-principle, applied to relate submersions and epimorphisms. It is shown that \(L\) is WI if and only if \(\nu(L)\) admits a trivial vector bundle extension over \(M\). It follows that, when \(M\) is contractible, the following conditions are equivalent: (1)~\(L\) is WI; (2)~\(\nu(L)\) extends over \(M\); and (3)~\(T(L)\) extends over \(M\) (thus \(L\) is parallelizable). More specific results are given when \(m=2\).NEWLINENEWLINEIn the rest of the paper, the authors consider only the case \(M=\mathbb{E}^n\), where they use the Hirsch-Smale classification of immersions, as well as tangential and normal characteristic classes.NEWLINENEWLINEFor \(n\geq2k+1\), they show that the following conditions are equivalent: (1)~any embedding \(L\to\mathbb{E}^n\) is WI; (2)~there is a WI embedding \(L\to\mathbb{E}^n\); and (3)~\(L\) is parallelizable. As a corollary, they get that any embedding of a \(3\)-manifold in \(\mathbb{E}^7\) is WI, and any embedding with trivial normal bundle of a \(7\)-manifold in \(\mathbb{E}^{15}\) is WI. When \(k\leq n-1\), it is proved that \(L\) is WI if and only if \(\nu(L)\) is trivial.NEWLINENEWLINEFor \(k\not\in\{3,7\}\), they prove that any parallelizable \(k\)-manifold is TWI. For \(k\in\{3,7\}\), it is proved that a \(k\)-manifold is TWI if and only if its connected components are parallelizable and have zero semicharasteristic modulo \(2\).NEWLINENEWLINEWhen \(n\geq2k+1\), it is shown that, if \(L\) is parallelizable, then any embedding \(L\to\mathbb{E}^n\) is SI. Moreover this also holds for \(n=2k+1\) if \(k\geq2\) and \(L\) is connected and TWI, or if \(L\) is open.NEWLINENEWLINEIn the case \(2+k\leq n\leq2k\), it is proved that \(L\) is SI if and only if it is WI and CI. It follows that the SI and WI properties are equivalent if \(m\in\{2,4,8\}\), and that any parallelizable \(4\)-manifold admits an SI embedding in \(\mathbb{E}^6\).NEWLINENEWLINENext the authors consider the particular case where \(L\) is a link in \(\mathbb{E}^3\). Now the use of Seifert surfaces plays an important role in the arguments. Let \(L_i\) denote the connected components of \(L\), and \(\text{lk}(L_i,L_j)\) the linking number between every \(L_i\) and \(L_j\). It is proved that \(L\) is SI if and only if \(\sum_{j\neq i}\text{lk}(L_i,L_j)=1\;\text{mod}\;2\). In particular, no knot in \(\mathbb{E}^3\) is SI, which surprisingly contradicts a previous claim of Miyoshi. Some of these ideas are generalized to the case \(k\in\{3,7\}\), obtaining e.g. that no embedding \(\mathbb{S}^k\to\mathbb{E}^{2k+1}\) is SI.NEWLINENEWLINEThose results are applied to characterize the proper submanifolds of \(\mathbb{E}^n\) that can be realized as leaves of foliations. For instance, \(\mathbb{S}^3\) can be realized as leaf of a foliation in \(\mathbb{E}^7\), but not in \(\mathbb{E}^5\) or \(\mathbb{E}^6\), which answers a question of \textit{E. Vogt} [Math. Ann. 296, No. 1, 159--178 (1993; Zbl 0816.57018)].NEWLINENEWLINEFinally, it is shown that these results about embeddings generalize to the category of analytic manifolds.