The cyclic bar construction and fundamental groups

We determine the 0-th Hochschild homology of the associative algebra of simplicial cochains valued in a PID: it consists of the ``finite-type" homotopy invariants of free loops, equivalently finite-type class functions on the fundamental group. One major motivation for this calculation is joint work in progress aiming to geometrically construct invariants of links in the 3-sphere as well as other $3$-manifolds, and to realize Milnor’s linking numbers as evaluations of 0-th Hochschild homology classes.