An algebraic model for the constant loops map
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908567560257536 |
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| author | Fernandez, Luis Rivera, Manuel Tradler, Thomas |
| author_facet | Fernandez, Luis Rivera, Manuel Tradler, Thomas |
| contents | For any simplicial complex $X$ with a total ordering of its vertices, one can construct a chain complex $\mathbb{L}_\bullet(X)$ generated by necklaces of simplices in $X$, which computes the homology of the free loop space of the geometric realization of $X$. Motivated by string topology, we describe two explicit chain maps $C_\bullet(X) \to \mathbb{L}_\bullet(X)$, where $C_\bullet(X)$ denotes the simplicial chains in $X$, lifting the homology map induced by embedding points in $|X|$ into constant loops in the free loop space of $|X|$. One of the maps has a convenient combinatorial description, while the other is described in terms of higher structure on $C_\bullet(X)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_18726 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An algebraic model for the constant loops map Fernandez, Luis Rivera, Manuel Tradler, Thomas Algebraic Topology 16E40, 57P10, 55P50 For any simplicial complex $X$ with a total ordering of its vertices, one can construct a chain complex $\mathbb{L}_\bullet(X)$ generated by necklaces of simplices in $X$, which computes the homology of the free loop space of the geometric realization of $X$. Motivated by string topology, we describe two explicit chain maps $C_\bullet(X) \to \mathbb{L}_\bullet(X)$, where $C_\bullet(X)$ denotes the simplicial chains in $X$, lifting the homology map induced by embedding points in $|X|$ into constant loops in the free loop space of $|X|$. One of the maps has a convenient combinatorial description, while the other is described in terms of higher structure on $C_\bullet(X)$. |
| title | An algebraic model for the constant loops map |
| topic | Algebraic Topology 16E40, 57P10, 55P50 |
| url | https://arxiv.org/abs/2411.18726 |