О вложениях четырехмерных многообразий в $\mathbf R^7$

We shall give a short survey on classification of embedings of 4-manifolds into $m$-dimensional Euclidean spaces. Such a classification for $m=7$ was known only for simply-connected manifolds (Boechat and Haefliger, 1970). The main result is a classification of embeddings of the product $S^1\times S^3$ into $R^7$. The group of such embeddings is isomorphic to
$$ Z\oplus Z\oplus Z_2 \quad\text{or}\quad Z\oplus Z\oplus Z_2\oplus Z_{12} $$
in the piecewise linear or smooth category, respectively. All the generators of this group will be explicitly constructed. Such constructions, based on Borromean rings and Whitehead link, are nice illustrations of higher-dimensional intuitive topology. The invariants allowing to obtain such a classification results are:
1) the Haefliger-Wu invariant, derived from the configuration space of pairs of distinct points, and
2) the Hudson-Habegger invariant, derived from analysis of self-intersections analogously to invariants, earlier introduced by Haefliger, Fenn, Rolfsen, Koschorke, Kirk, Saito and Levine.