Block-Decomposition for 3-Parameter Persistence Modules
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918021955584000 |
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| author | Yi, Siheng |
| author_facet | Yi, Siheng |
| contents | In 2020, Cochoy and Oudot got the necessary and sufficient condition of the block-decomposition of 2-parameter persistence modules $\mathbb{R}^2 \to \textbf{Vec}_{\Bbbk}$. And in 2024, Lebovici, Lerch and Oudot resolve the problem of block-decomposability for multi-parameter persistence modules. Following the approach of Cochoy and Oudot's proof of block-decomposability for 2-parameter persistence modules, we rediscuss the necessary and sufficient conditions for the block decomposition of the 3-parameter persistence modules $\mathbb{R}^3 \to \textbf{Vec}_{\Bbbk}$. Our most important contribution is to generalize the strong exactness of 2-parameter persistence modules to the case of 3-parameter persistence modules. What's more, the generalized method allows us to understand why there is no block decomposition in general persistence modules to some extent. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_08391 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Block-Decomposition for 3-Parameter Persistence Modules Yi, Siheng Algebraic Topology Representation Theory In 2020, Cochoy and Oudot got the necessary and sufficient condition of the block-decomposition of 2-parameter persistence modules $\mathbb{R}^2 \to \textbf{Vec}_{\Bbbk}$. And in 2024, Lebovici, Lerch and Oudot resolve the problem of block-decomposability for multi-parameter persistence modules. Following the approach of Cochoy and Oudot's proof of block-decomposability for 2-parameter persistence modules, we rediscuss the necessary and sufficient conditions for the block decomposition of the 3-parameter persistence modules $\mathbb{R}^3 \to \textbf{Vec}_{\Bbbk}$. Our most important contribution is to generalize the strong exactness of 2-parameter persistence modules to the case of 3-parameter persistence modules. What's more, the generalized method allows us to understand why there is no block decomposition in general persistence modules to some extent. |
| title | Block-Decomposition for 3-Parameter Persistence Modules |
| topic | Algebraic Topology Representation Theory |
| url | https://arxiv.org/abs/2505.08391 |