Embedding of \(k\)-orientable manifolds (Q1178425)

Let \(M\) be a closed, \(k\)-connected \(n\)-manifold, and \(x_0\in M\), \(M_0 = M - x_0\). In the report, the author proves: Theorem 1. Suppose \(M\) and \(M_0\) as above. If \(k\geq 1\), \(0\leq j\leq 2k\) and \(n\geq j+k+2\), then \(M\) embeds into \(\mathbb{R}^{2n-j-1}\) if and only if \(M_0\) immerses \(\mathbb{R}^{2n-j-2}\). Theorem 2. Let \(M\) and \(M_0\) be as in Theorem 1, and \(M\) be \((k+1)\)-orientable. If \(k\geq 1\) and \(n\geq 3k+4\), then \(M\) embeds into \(\mathbb{R}^{2n-2k-1}\) if and only if \(M_0\) immerses \(\mathbb{R}^{2n-2k-2}\). It is easily seen that the above results extend those of \textit{J. C. Becker} and \textit{H. H. Glover} [Proc. Am. Math. Soc. 27, 405--410 (1971; Zbl 0207.22402)] and of \textit{K. R. Ferland} [Mich. Math. J. 21, 253--256 (1974; Zbl 0298.57016)]. The argument is that the embeddings in these papers must be in the stable range, but this need not be. The author obtains the following corollary from Theorem 2: Corollary. Let \(M\) be a closed, \(k\)-connected \(n\)-manifold, and \(M\) be \(S\)-parallelizable (i.e. \(TM\oplus\Sigma^1\) is the trivial bundle). If \(n\geq 3k+4\), then \(M\) can be embedded into \(\mathbb{R}^{2n-2k-1}\).