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

TL;DR

This paper computes the cech homotopy groups of a shrinking wedge of spheres, revealing their structure as direct sums of homotopy groups of spheres and analyzing the canonical homomorphism's properties.

Contribution

It provides explicit calculations of cech homotopy groups for the infinite earring space and investigates the properties of the canonical homomorphism to these groups.

Findings

cech homotopy groups are isomorphic to direct sums of sphere homotopy groups.

cech homotopy groups are computed explicitly for all n,m.

The canonical homomorphism is a split epimorphism for nm-1.

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 $n>2m-1$.