An algebraic model for the constant loops map

Abstract

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 $C_{\bullet }(X)$.

21 pages, typos corrected