On the generalized Massey–Rolfsen invariant for link maps

A. Skopenkov¹

¹

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

Résumé

For $K=K_1\bigsqcup ...\bigsqcup K_s$ and a link map $f:K\to {ℝ}^m$ let $K^{\sim }={\bigsqcup }_{ij}{K}_i\times K_j$, 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 $\alpha (f)\in {\pi }^{m-1}(K)$ 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 ${\pi }^{m-1}(K^{\sim })$. If $K_1,...,K_s$ are closed highly homologically connected manifolds of dimension $p_1,...,p_s$ (in particular, homology spheres), then ${\pi }^{m-1}(K^{\sim })\cong {\oplus }_{ij}{\pi }_{p_i+p_j-m+1}^S$.

Détail

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

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