Abstract: 
We define an isotopy invariant of embeddings $N\to {{\fam \msyfam \relax R}}^m$ 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 us to obtain new completeness results for the $\alpha $-invariant and the following estimation of isotopy classes of embeddings. {\fam \itfam \tenit In the piecewise-linear category, for a $(3n-2m+2)$-connected $n$-manifold $N$ with ${(4n+5)/3}\le m\le {(3n+2)/2}$, each preimage of the $\alpha $-invariant injects into a quotient of $H_{3n-2m+3}(N)$, where the coefficients are ${{\fam \msyfam \relax Z}}$ for $m-n$ odd and ${{\fam \msyfam \relax Z}}_2$ for $m-n$ even.\/}

1. Department of Differential Geometry
   Faculty of Mechanics and Mathematics
   Moscow State University
   Moscow 119992, Russia
   and
   Independent University of Moscow
   B. Vlasyevskiy, 11, Moscow 119002, Russia