Note on the Embedding of Manifolds in Euclidean Space

Abstract

M. Hirsch and independently H. Glover have shown that a closed $k$-connected smooth $n$-manifold $M$ embeds in ${R^{2n - j}}$ if ${M_0}$ immerses in ${R^{2n - j - 1}},j \leqq 2k$ and $2j \leqq n - 3$. Here ${M_0}$ denotes $M$ minus the interior of a smooth disk. In this note we prove the converse and show also that the isotopy classes of embeddings of $M$ in ${R^{2n - j}}$ are in one-one correspondence with the regular homotopy classes of immersions of ${M_0}$ in ${R^{2n - j - 1}},j \leqq 2k - 1$ and $2j \leqq n - 4$.