Transformation Groups Seminar
Speaker: Matthias Franz (Western)
"The Szczarba map and the cubical cobar construction"
Time: 09:30 - 10:30
Room: MC 108
Let $X$ be a simplicial set and $G$ a simplicial group. Any group morphism from the Kan loop group $\Omega X$ to $G$ is determined by a twisting function $\tau\colon X\to G$. In 1961, Szczarba gave an explicit construction of a twisting cochain $t\colon C(X)\to C(G)$ out of a twisting function $X\to G$. Such a twisting cochain induces a multiplicative map from the cobar construction $\boldsymbol{\Omega}\,C(X)$ to $C(G)$.
Recently I proved that the map induced by Szczarba's twisting cochain is also comultiplicative; the coproduct on $\boldsymbol{\Omega}\,C(X)$ is defined in terms of homotopy Gerstenhaber operations on $X$. Shortly afterwards, Minichiello--Rivera--Zeinalian gave a conceptual explanation of this fact, based on the idea of triangulating the cubical cobar construction of $X$. In this talk I want to elucidate the properties of Szczarba's twisting cochain that make this construction possible.