Abstract: 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.

Abstract: In 1992 Jeffrey and Weitsman successfully employed half-density quantization to the moduli space of flat connections and the quantization was applied to produce 3-manifold invariants for compact oriented 3-manifolds. These invariants have impetus in the asymptotic expansion of the Witten–Reshetikhin–Turaev (WRT) invariants, which Witten conjectured that the expansion depended on invariants of the flat connections: the Chern-Simons invariant, the Reidmeister torsion and spectral flow. Using a Chern-Simons line bundle paired with "generalized-half-density" bundle, the first two of these connection invariants were made to appear. A compact oriented 3-manifold admits a Heegaard decomposition: defining these invariants was reduced to a pairing of sections of these two bundles on two handlebodies and their mutual bounding genus-g surface. This was done in a covariant constant fashion along one special non-smooth leaf of a real "polarization" of this moduli space. In this talk, we will look at different formulations of the moduli space of flat connections required for the quantization, recall the prequantum setup, and focus on the polarization to which adapted half-densities and half-forms exist.