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The Čech homotopy groups of a shrinking wedge of spheres

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 $n,m\geq 2$, we prove that $\checkπ_n(\mathbb{E}_m)$ is isomorphic to a direct sum of countable powers of homotopy groups of spheres: $\bigoplus_{1\leq j\leq \frac{n-1}{m-1}}\left(π_{n}(S^{mj-j+1})\right)^{\mathbb{N}}$. Equipped with this isomorphism and infinite-sum algebra, we also construct new elements of $π_n(\mathbb{E}_m)$ with a view toward characterizing the image of the canonical homomorphism $Ψ_{n}:π_n(\mathbb{E}_m)\to \checkπ_{n}(\mathbb{E}_m)$. We prove that $Ψ_{n}$ is a split epimorphism when $n\leq 2m-1$ and we identify a candidate for the image of $Ψ_n$ when $n>2m-1$.