Manchester Geometry Seminar 2001/2002

We shall give a survey on embeddings of 4-manifolds into Euclidean spaces. Since 1970 until now a complete classification of embeddings of 4-manifolds into R⁷ was known only for the simply-connected case. The main result of the talk is that the group of embeddings of S¹× S³ into R⁷ is isomorphic to Z+Z+Z/2. All the representatives of isotopy classes of embeddings will be explicitly constructed, using Stiefel manifolds, Borromean rings and Whitehead link. The Haefliger-Wu and the Hudson-Habegger invariants, allowing to distinguish embeddings, will be defined.