Classification of embeddings below the metastable dimension

Abstract

We develop a new approach to the classical problem on isotopy classification of embeddings of manifolds into Euclidean spaces. This approach involves studying of a new embedding invariant, of almost-embeddings and of smoothing, as well as explicit constructions of embeddings. Using this approach we obtain complete concrete classification results below the metastable dimension range, i.e. where the configuration spaces invariant of Haefliger-Wu is incomplete. Note that all known complete concrete classification results, except for the Haefliger classification of links and smooth knots, can be obtained using the Haefliger-Wu invariant. More precisely, we classify embeddings S^p x S^{2l-1} -> R^{3l+p} for p<l in terms of homotopy groups of Stiefel manifolds (up to minor indeterminancies for p>1 and for the smooth category). A particular case states that the set of piecewise-linear isotopy classes of piecewise-linear embeddings (or, equivalently, the set of almost smooth isotopy classes of smooth embeddings) S^1 x S^3 -> R^7 has a geometrically defined group structure, and with this group structure is isomorphic to Z + Z + Z_2. We exhibit an example disproving the conjecture proposed by Viro and others on the completeness of the multiple Haefliger-Wu invariant for classification of PL embeddings of connected manifolds in codimension at least 3.