Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The cyclic bar construction and fundamental groups

Nir Gadish

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.

The cyclic bar construction and fundamental groups

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 -manifolds, and to realize Milnor's linking numbers as evaluations of 0-th Hochschild homology classes.

Paper Structure

This paper contains 3 sections, 11 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Associative algebras
  3. Letter braiding invariants of loops

Key Result

Theorem 1.1

Let $Y$ be a connected pointed simplicial set, with finitely generated fundamental group $\pi_1(Y)$. Let $C^*_{\Delta}(Y;R)$ denote the $R$-valued simplicial cochains algebra with its standard Alexander--Whitney cup product, and $\operatorname{Cyc}_{\Delta}(Y;R)$ the associated cyclic Bar complex (a where the target is the module all finite type functions $\pi_1(Y)\to R$ that are furthermore invar

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2: Cyclic Bar construction
  • Example 2.3: Linear Bar construction
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Lemma 2.8
  • ...and 18 more