The cyclic bar construction and fundamental groups

TL;DR

This paper computes the 0-th Hochschild homology of simplicial cochains valued in a PID, linking it to finite-type invariants of free loops and fundamental groups, with applications to link invariants and 3-manifold topology.

Contribution

It provides a new calculation connecting Hochschild homology with finite-type invariants and fundamental groups, advancing geometric approaches to link and 3-manifold invariants.

Findings

0-th Hochschild homology characterized by finite-type homotopy invariants

Establishes a link between Hochschild classes and Milnor's linking numbers

Potential applications to geometric link invariants in 3-manifolds

Abstract

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.