On the generalized Massey–Rolfsen invariant for link maps

A. Skopenkov

Fundamenta Mathematicae, Tome 165 (2000) no. 1, pp. 1-15

Résumé

For $K = K_1⊔...⊔K_s$ and a link map $f:K → ℝ^m$ let $K^∼ = ⊔_{i j} K_i × K_j$, define a map $f^∼ : K^∼ → S^{m - 1}$ by $f^∼(x, y) = (fx - fy)/|fx - fy|$ and a (generalized) Massey-Rolfsen invariant $α(f) ∈ π^{m - 1}(K)$ 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 $f:K → ℝ^m$ up to link concordance to $π^{m - 1}(K^∼)$. If $K_1,...,K_s$ are closed highly homologically connected manifolds of dimension $p_1,...,p_s$ (in particular, homology spheres), then $π^{m-1}(K^∼)≅⊕_{i j} π^S_{p_i + p_j - m + 1}$.

Détail

DOI : 10.4064/fm-165-1-1-15

Keywords: deleted product, Massey-Rolfsen invariant, link maps, link homotopy, stable homotopy group, double suspension, codimension two, highly connected manifolds