The Čech homotopy groups of a shrinking wedge of spheres
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917195320131584 |
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| author | Brazas, Jeremy |
| author_facet | Brazas, Jeremy |
| contents | 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 $n>2m-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15730 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Čech homotopy groups of a shrinking wedge of spheres Brazas, Jeremy Algebraic Topology 55Q07, 55Q52, 55Q20, 55Q15, 08A65 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 $n>2m-1$. |
| title | The Čech homotopy groups of a shrinking wedge of spheres |
| topic | Algebraic Topology 55Q07, 55Q52, 55Q20, 55Q15, 08A65 |
| url | https://arxiv.org/abs/2501.15730 |