The bitwisted Cartesian model for the free loop fibration

TL;DR

This paper introduces a novel bitwisted Cartesian model for the free loop fibration using truncating twisting functions, providing explicit algebraic structures and models that connect cubical and simplicial sets.

Contribution

It develops a new bitwisted Cartesian product framework and constructs explicit models and diagonals for the free loop fibration, linking cubical and simplicial chain complexes.

Findings

The chain complex of the bitwisted product matches the Hochschild chain complex.

Constructed polytopes $F_n$ with explicit diagonals.

Established an algebra isomorphism for the cohomology of free loop spaces.

Abstract

Using the notion of truncating twisting function from a simplicial set to a cubical set a special, bitwisted, Cartesian product of these sets is defined. For the universal truncating twisting function, the (co)chain complex of the corresponding bitwisted Cartesian product agrees with the standard Cartier (Hochschild) chain complex of the simplicial (co)chains. The modelling polytopes $F_n$ are constructed. An explicit diagonal on $F_n$ is defined and a multiplicative model for the free loop fibration $\Omega Y\to \Lambda Y\to Y$ is obtained. As an application we establish an algebra isomorphism $H^*(\Lambda Y;\mathbb{Z}) \approx S(U)\otimes \Lambda(s^{_{-1}}U)$ for the polynomial cohomology algebra $H^*(Y;\mathbb{Z})=S(U).$