The bitwisted Cartesian model for the free loop fibration

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 $H^*(Y;\mathbb{Z})=S(U).$