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Abstract

We define an isotopy invariant of embeddings $N\to {{\mathbb R}}^m$ 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. 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 ${{\mathbb Z}}$ for $m-n$ odd and ${{\mathbb Z}}_2$ for $m-n$ even.