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 ΩX to G is determined by a twisting function τ:X→G. In 1961, Szczarba gave an explicit construction of a twisting cochain t:C(X)→C(G) out of a twisting function X→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.