Minimal Morse functions on a pair of manifolds (Q1332034)

The existence theorem for a minimal Morse function on a pair of manifolds \((M^n,N^k)\) is proved: Theorem. Assume that a manifold \(N^k\) is imbedded in a manifold \(M^n\) so that \(k \geq 6\), \(n-k \geq 3\), and \(\pi_1(M) = \pi_1(N) = 0\). Then, for sufficiently large \(m\), there exists an embedding \(M\) in \(\mathbb{R}^{m+1}\) such that the imbeddings \(N\) and \(M\) in \(\mathbb{R}^{m+1}\) are imbeddings with critical levels under which the level function on \(N\) is a bound for the level function on \(M\). The number of handles with each index \(\lambda\) is minimal on both manifolds \(N\) and \(M\) and is equal to \(m_\lambda = \mu (H_\lambda (M^n)) + \mu (\text{Tors} H_{\lambda-1} (M^n))\) on the manifold \(M\), and to \(n_\lambda = \mu (H_\lambda (N^k))+\mu(\text{Tors} H_{\lambda-1} (M^n))\) on the manifold \(N\), where \(\mu (H)\) is the minimal number of generators of the group \(H\). In the Diff category, we have the following Theorem. Let \((M^n, N^k)\) be a pair of smooth connected manifolds of the class \(C^\infty\), and \(k \geq 6\), \(n-k \geq 3\). Then there exists a minimal Morse function \(M\) such that its restriction to \(N\) is also a minimal Morse function. The proof is similar to the proof in the PL-category.