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arXiv · 2504.11223

The Simplicial Loop Space of a Simplicial Complex

Gregory Lupton·Jonathan Scott
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Abstract

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 $X$.

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Gregory Lupton, Jonathan Scott. 2025-07-16. The Simplicial Loop Space of a Simplicial Complex. https://arxiv.org/abs/2504.11223

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