An algebraic model for the constant loops map

TL;DR

The paper constructs an algebraic model $\mathbb{L}_\bullet(X)$, built from necklaces of simplices, that computes the homology of the free loop space $L|X|$ and provides two explicit chain-level lifts $\rho$ (combinatorial) and $\chi$ (higher-structure-based) of the constant-loops embedding. It establishes the equivalence of these lifts on homology with the geometric embedding $|X|\hookrightarrow L|X|$ and proves $\rho$ and $\chi$ are chain-homotopic. Beyond the primary lift, the authors develop an alternate model via a dg category $\mathbb{P}_\bullet(X)$ equipped with a coproduct $\nabla_0$, antipode $S$, and chain homotopy $\nabla_1$, enabling a detailed construction of $\chi$ and clarifying the role of higher algebraic structures (e.g., $E_2$ and $E_3$-structures) in string topology. The work connects to coHochschild/Hochschild theories, Adams cobar duality, and string topology operations such as the Goresky–Hingston coproduct, offering explicit chain-level tools and Calabi–Yau perspectives for calculations on loop spaces.

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 $C_\bullet(X)$.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Theorem 2.4
• Remark 2.5
• Remark 2.6
• Example 3.1