The Simplicial Loop Space of a Simplicial Complex
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913943977459712 |
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| author | Lupton, Gregory Scott, Jonathan |
| author_facet | Lupton, Gregory Scott, Jonathan |
| contents | Given a simplicial complex $X$, we construct a simplicial complex $ΩX$ that may be regarded as a combinatorial version of the based loop space of a topological space. Our construction explicitly describes the simplices of $ΩX$ directly in terms of the simplices of $X$. Working at a purely combinatorial level, we show two main results that confirm the (combinatorial) algebraic topology of our $ΩX$ behaves like that of the topological based loop space. Whereas our $ΩX$ is generally a disconnected simplical complex, each component of $ΩX$ has the same edge group, up to isomorphism. We show an isomorphism between the edge group of $ΩX$ and the combinatorial second homotopy group of $X$ as it has been defined in separate work (arxiv:2503.23651). Finally, we enter the topological setting and, relying on prior work of Stone, show a homotopy equivalence between the spatial realization of our $ΩX$ and the based loop space of the spatial realization of $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11223 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Simplicial Loop Space of a Simplicial Complex Lupton, Gregory Scott, Jonathan Algebraic Topology 55U10 (Primary), 55P35, 05E45 (Secondary) Given a simplicial complex $X$, we construct a simplicial complex $ΩX$ that may be regarded as a combinatorial version of the based loop space of a topological space. Our construction explicitly describes the simplices of $ΩX$ directly in terms of the simplices of $X$. Working at a purely combinatorial level, we show two main results that confirm the (combinatorial) algebraic topology of our $ΩX$ behaves like that of the topological based loop space. Whereas our $ΩX$ is generally a disconnected simplical complex, each component of $ΩX$ has the same edge group, up to isomorphism. We show an isomorphism between the edge group of $ΩX$ and the combinatorial second homotopy group of $X$ as it has been defined in separate work (arxiv:2503.23651). Finally, we enter the topological setting and, relying on prior work of Stone, show a homotopy equivalence between the spatial realization of our $ΩX$ and the based loop space of the spatial realization of $X$. |
| title | The Simplicial Loop Space of a Simplicial Complex |
| topic | Algebraic Topology 55U10 (Primary), 55P35, 05E45 (Secondary) |
| url | https://arxiv.org/abs/2504.11223 |