Obstructions to embedding disks. II: A proof of a conjecture of Hudson
This paper is concerned with the problem of representing an element of \(\pi_ n(W,\partial W)\) by a properly imbedded disk - W is a PL-manifold of dimension m. J. F. P. Husdon defines an obstruction in \(H_{2n- m+1}(W,\partial W;G)\) where \(G={\mathbb{Z}}\) or \({\mathbb{Z}}/2\) as m-n is odd or even. In certain cases Hudson proves that the vanishing of this obstruction is sufficient for the existence of an imbedding. For example, \(m\geq n+3\) and (W,\(\partial W) (2n-m)-\)connected are certainly necessary conditions, but Hudson also requires m-n even and \(\pi_ 1(W)=\pi_ 1(\partial W)=0.\) In the present paper these latter conditions are eliminated and the obstructions are generalized to encompass the problem of representing several elements of \(\pi_ n(W,\partial W)\) by disjointly imbedded disks. In carrying out this project intersection and self-intersection invariants are introduced in the homology of \((\tilde W,\partial\tilde W)\), the universal covering pair of (W,\(\partial W)\).
representation of elements in relative homotopy groups
scientific article; zbMATH DE number 3872292