A combinatorial model for the free loop fibration

TL;DR

This paper develops a combinatorial model for the free loop fibration using closed necklical sets, providing a new way to understand free loop spaces through simplicial and chain complex constructions.

Contribution

It introduces the concept of closed necklical sets to model free loop fibrations functorially, linking geometric realizations to homotopy types and chain complexes.

Findings

The geometric realization of the closed necklical set models the free loop space homotopy equivalently.

The chain complex associated generalizes the coHochschild chain complex.

Provides a new combinatorial framework for studying free loop fibrations.

Abstract

We introduce the abstract notion of a closed necklical set in order to describe a functorial combinatorial model of the free loop fibration $\Omega Y\rightarrow \Lambda Y\rightarrow Y$ over the geometric realization $Y=|X|$ of a path connected simplicial set $X.$ In particular, to any path connected simplicial set $X$ we associate a closed necklical set $\widehat{\mathbf{\Lambda}}X$ such that its geometric realization $|\widehat{\mathbf{\Lambda}}X|$, a space built out of gluing "freehedrical" and "cubical" cells, is homotopy equivalent to the free loop space $\Lambda Y$ and the differential graded module of chains $C_*(\widehat{\mathbf{\Lambda}}X)$ generalizes the coHochschild chain complex of the chain coalgebra $C_\ast(X).$