An algebraic model for the constant loops map

TL;DR

This paper introduces an algebraic model using necklaces of simplices to compute the homology of free loop spaces, providing explicit chain maps that relate simplicial chains to constant loops, inspired by string topology.

Contribution

It presents a novel algebraic framework and explicit chain maps connecting simplicial chains to the homology of free loop spaces, advancing the understanding of string topology.

Findings

Constructed chain complex $ L_ullet(X)$ for necklaces in $X$

Defined explicit chain maps lifting the constant loops map

Provided combinatorial and higher-structure descriptions of the maps

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 $C_\bullet(X)$.