Embeddings of $k$-connected $n$-manifolds into $\mathbb {R}^{2n-k-1}$

Abstract:

We obtain estimations for isotopy classes of embeddings of closed $k$-connected $n$-manifolds into $\mathbb {R}^{2n-k-1}$ for $n\ge 2k+6$ and $k\ge 0$. This is done in terms of an exact sequence involving the Whitney invariants and an explicitly constructed action of $H_{k+1}(N;\mathbb {Z}_{2})$ on the set of embeddings. The proof involves a reduction to the classification of embeddings of a punctured manifold and uses the parametric connected sum of embeddings.

Corollary. Suppose that $N$ is a closed almost parallelizable $k$-connected $n$-manifold and $n\ge 2k+6\ge 8$. Then the set of isotopy classes of embeddings $N\to \mathbb {R}^{2n-k-1}$ is in 1–1 correspondence with $H_{k+2}(N;\mathbb {Z} _{2})$ for $n-k=4s+1$.