Homotopy properties of subsets of Euclidean spaces (Q500932)

When considering homotopy properties of locally complicated spaces, few of the classical results may be of use as local anomalies may induce enormous complications to the induced invariants. In the case of the fundamental group, one usually assumes come kind of local connectedness or the property of being homotopically Hausdorff in order to obtain reasonable results. This paper assumes the generalization of the mentioned properties to the higher dimensional case and presents the corresponding generalizations of known results to higher homotopy groups. Amongst other results, the authors prove that every subset of \(\mathbb{R}^{n+1}\) is \(n\)-homotopically Hausdorff. Furthermore, they provide local connectivity conditions on a space \(X\) under which \(\pi_n(X)\) is free Abelian.