The \v{C}ech homotopy groups of a shrinking wedge of spheres

Abstract

We compute the \v{C}ech homotopy groups of the $m$-dimensional infinite earring space $\mathbb{E}_m$, i.e. a shrinking wedge of $m$-spheres. In particular, for all $n,m\geq 2$, we prove that $\check{\pi}_n(\mathbb{E}_m)$ is isomorphic to a direct sum of countable powers of homotopy groups of spheres: $\bigoplus_{1\leq j\leq \frac{n-1}{m-1}}\left(\pi_{n}(S^{mj-j+1})\right)^{\mathbb{N}}$. 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 $n\leq 2m-1$ and we identify a candidate for the image of $\Psi_n$ when $n>2m-1$.