A new invariant and parametric connected sum of embeddings

Abstract

We define an isotopy invariant of embeddings N -> R^m of manifolds into Euclidean space. This invariant together with the \alpha-invariant of Haefliger-Wu is complete in the dimension range where the \alpha-invariant could be incomplete. We also define parametric connected sum of certain embeddings (analogous to surgery). This allows to obtain new completeness results for the \alpha-invariant and the following estimation of isotopy classes of embeddings. For the piecewise-linear category, a (3n-2m+2)-connected n-manifold N and (4n+4)/3 < m < (3n+3)/2 each preimage of \alpha-invariant injects into a quotient of H_{3n-2m+3}(N), where the coefficients are Z for m-n odd and Z_2 for m-n even.