Higher homotopy wild sets
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908384325795840 |
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| author | Brazas, Jeremy Mitra, Atish |
| author_facet | Brazas, Jeremy Mitra, Atish |
| contents | The $π_n$-wild set $\mathbf{w}_{n}(X)$ of a topological space $X$ is the subspace of $X$ consisting of the points at which there exists a shrinking sequence of essential based maps $S^n\to X$. In this paper, we show that the homotopy type of $\mathbf{w}_{n}(X)$ is a homotopy invariant of $X$ and, in analogy to the known one-dimensional case, we show that for certain $n$-dimensional $π_n$-shape injective metric spaces, the homeomorphism type of $\mathbf{w}_{n}(X)$ is a homotopy invariant of $X$. We also prove that the $π_n$-wild set of a Peano continuum can be homeomorphic to any compact metric space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_23665 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher homotopy wild sets Brazas, Jeremy Mitra, Atish Algebraic Topology General Topology 54F15, 55Q52, 55Q35 The $π_n$-wild set $\mathbf{w}_{n}(X)$ of a topological space $X$ is the subspace of $X$ consisting of the points at which there exists a shrinking sequence of essential based maps $S^n\to X$. In this paper, we show that the homotopy type of $\mathbf{w}_{n}(X)$ is a homotopy invariant of $X$ and, in analogy to the known one-dimensional case, we show that for certain $n$-dimensional $π_n$-shape injective metric spaces, the homeomorphism type of $\mathbf{w}_{n}(X)$ is a homotopy invariant of $X$. We also prove that the $π_n$-wild set of a Peano continuum can be homeomorphic to any compact metric space. |
| title | Higher homotopy wild sets |
| topic | Algebraic Topology General Topology 54F15, 55Q52, 55Q35 |
| url | https://arxiv.org/abs/2505.23665 |