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For $K=K_1\bigsqcup ...\bigsqcup K_s$ and a link map $f:K\to {ℝ}^m$ let $K^{\sim }={\bigsqcup }_{i<j}{K}_i\times K_j$, define a map $f^{\sim }:K^{\sim }\to S^{m-1}$ by $f^{\sim }(x,y)=(fx-fy)/|fx-fy|$ and a (generalized) Massey-Rolfsen invariant $\alpha \left(f\right)\in {\pi }^{m-1}\left(K\right)$ to be the homotopy class of $f^{\sim }$. We prove that for a polyhedron K of dimension ≤ m - 2 under certain (weakened metastable) dimension restrictions, α is an onto or a 1 - 1 map from the set of link maps $f:K\to {ℝ}^m$ up to link concordance to ${\pi }^{m-1}\left(K^{\sim }\right)$. If $K_1,...,K_s$ are closed highly homologically connected manifolds of dimension $p_1,...,p_s$ (in particular, homology spheres), then ${\pi }^{m-1}\left(K^{\sim }\right)\cong {\oplus }_{i<j}{\pi }_{p_i+p_j-m+1}^S$.

We define an isotopy invariant of embeddings $N\to {ℝ}^m$ of manifolds into Euclidean space. This invariant together with the α-invariant of Haefliger-Wu is complete in the dimension range where the α-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 α-invariant and the following estimation of isotopy classes of embeddings. In the piecewise-linear category, for a (3n-2m+2)-connected...