The unstable homotopy groups of 2-cell complexes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915008449871872 |
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| author | Zhu, Zhongjian |
| author_facet | Zhu, Zhongjian |
| contents | In this paper, we develop the new method, initiated by B. Gray (1972), to compute the unstable homotopy groups of the mapping cone, especially for $2$-cell complex $X=S^m\cup_α e^{n}$. By Gray's work mentioned above or the traditional method given by I.M.James (1957) which were widely used in previous related work to compute $π_{i}(X)$, the dimension $i\leq 2n+m-4$. By our method, we can compute $π_{i}(X)$ for $i>2n+m-4$. We use this different technique to generalize J.Wu's work, at Mem. of AMS, on homotopy groups of mod $2$ Moore spaces to higher dimensional homotopy groups of mod $2^r$ Moore spaces $P^{n}(2^r)$ for all $r\geq 1$. This practice shows that the technique given here is a new general method to compute the unstable homotopy groups of CW complexes with higher dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20416 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The unstable homotopy groups of 2-cell complexes Zhu, Zhongjian Algebraic Topology In this paper, we develop the new method, initiated by B. Gray (1972), to compute the unstable homotopy groups of the mapping cone, especially for $2$-cell complex $X=S^m\cup_α e^{n}$. By Gray's work mentioned above or the traditional method given by I.M.James (1957) which were widely used in previous related work to compute $π_{i}(X)$, the dimension $i\leq 2n+m-4$. By our method, we can compute $π_{i}(X)$ for $i>2n+m-4$. We use this different technique to generalize J.Wu's work, at Mem. of AMS, on homotopy groups of mod $2$ Moore spaces to higher dimensional homotopy groups of mod $2^r$ Moore spaces $P^{n}(2^r)$ for all $r\geq 1$. This practice shows that the technique given here is a new general method to compute the unstable homotopy groups of CW complexes with higher dimension. |
| title | The unstable homotopy groups of 2-cell complexes |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2410.20416 |