The Čech homotopy groups of a shrinking wedge of spheres

Abstract

We compute the Čech homotopy groups of the $m$-dimensional infinite earring space ${\mathbb{E}}_m$, i.e. a shrinking wedge of $m$-spheres. In particular, for all $n,m\ge 2$, we prove that ${\check{\pi }}_n({\mathbb{E}}_m)$ is isomorphic to a direct sum of countable powers of homotopy groups of spheres: ${⨁}_{1\le j\le \frac{n-1}{m-1}}{\left({\pi }_n(S^{mj-j+1})\right)}^{\mathbb{N}}$. Equipped with this isomorphism and infinite-sum algebra, we also construct new elements of ${\pi }_n({\mathbb{E}}_m)$ with a view toward characterizing the image of the canonical homomorphism ${\Psi }_n:{\pi }_n({\mathbb{E}}_m)\to {\check{\pi }}_n({\mathbb{E}}_m)$. We prove that ${\Psi }_n$ is a split epimorphism when $n\le 2m-1$ and we identify a candidate for the image of ${\Psi }_n$ when $n>2m-1$.

16 pages