On the generalized Massey–Rolfsen invariant for link maps

For K = K1t . . .tKs and a link map f : K → R letK = ⊔ i<j Ki×Kj , define a map f :K → Sm−1 by f(x, y) = (fx− fy)/|fx− fy| and a (generalized) Massey– Rolfsen invariant α(f) ∈ πm−1(K) to be the homotopy class of f . 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 → R up to link concordance to πm−1(K). If K1, . . . ,Ks are closed highly homologically connected manifolds of dimension p1, . . . , ps (in particular, homology spheres), then πm−1(K) ∼= ⊕ i<j π S pi+pj−m+1.

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