Homotopy groups of compact Hausdorff spaces with trivial shape

Abstract:

Given a collection $\{ {\pi _n}:n = 1,2, \cdots \}$ of countable groups such that ${\pi _i}$ is abelian and admits ${\pi _1}$ as a group of operators for $i \geqq 2$, we construct here an arcwise connected compact metric space of trivial shape whose $j$th homotopy group is isomorphic to ${\pi _j}$ for $j = 1,2, \cdots$ . The isomorphisms preserve the action of the first group on the higher groups. Thus, the homotopy groups of a compact metric space of trivial shape may be quite arbitrary.