A combinatorial model for the free loop fibration

Abstract

We introduce the abstract notion of a closed necklical set in order to describe a functorial combinatorial model of the free loop fibration $\Omega Y\to \Lambda Y\to Y$ 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 $\hat{\mathbf{\Lambda }}X$ such that its geometric realization $|\hat{\mathbf{\Lambda }}X|$, 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 $C_{\ast }(\hat{\mathbf{\Lambda }}X)$ generalizes the coHochschild chain complex of the chain coalgebra $C_{\ast }(X).$

Made minor revisions. To appear in Bulletin of the London Mathematical Society

Cited by in corpus (2)

• Hochschild lattices and shuffle lattices
• Hochschild polytopes