Discrete Morse theory for open complexes
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911739906359296 |
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| author | Knudson, Kevin P. Scoville, Nicholas A. |
| author_facet | Knudson, Kevin P. Scoville, Nicholas A. |
| contents | We develop a discrete Morse theory for open simplicial complexes $K=X\setminus T$ where $X$ is a simplicial complex and $T$ a subcomplex of $X$. A discrete Morse function $f$ on $K$ gives rise to a discrete Morse function on the order complex $S_K$ of $K$, and the topology change determined by $f$ on $K$ can be understood by analyzing the topology change determined by the discrete Morse function on $S_K$. This topology change is given by a structure theorem on the level subcomplexes of $S_K$. Finally, we show that the Borel-Moore homology of $K$, a homology theory for locally compact spaces, is isomorphic to the homology induced by a gradient vector field on $K$ and deduce corresponding weak Morse inequalities. The gradient vector field on $K$ provides a novel alternative to compute Borel-Moore homology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12116 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discrete Morse theory for open complexes Knudson, Kevin P. Scoville, Nicholas A. Algebraic Topology 57Q70, 55U10, 55N35 We develop a discrete Morse theory for open simplicial complexes $K=X\setminus T$ where $X$ is a simplicial complex and $T$ a subcomplex of $X$. A discrete Morse function $f$ on $K$ gives rise to a discrete Morse function on the order complex $S_K$ of $K$, and the topology change determined by $f$ on $K$ can be understood by analyzing the topology change determined by the discrete Morse function on $S_K$. This topology change is given by a structure theorem on the level subcomplexes of $S_K$. Finally, we show that the Borel-Moore homology of $K$, a homology theory for locally compact spaces, is isomorphic to the homology induced by a gradient vector field on $K$ and deduce corresponding weak Morse inequalities. The gradient vector field on $K$ provides a novel alternative to compute Borel-Moore homology. |
| title | Discrete Morse theory for open complexes |
| topic | Algebraic Topology 57Q70, 55U10, 55N35 |
| url | https://arxiv.org/abs/2402.12116 |