Natural transformations relating homotopy and singular homology functors

TL;DR

This paper constructs natural transformations connecting algebraic and topological invariants, specifically relating certain quotients of fundamental group algebras to singular homology groups, and links these to Beilinson's work.

Contribution

It introduces a family of natural transformations between algebraic and homological functors on topological spaces with marked points, extending and relating to Beilinson's equivalence.

Findings

Established natural transformations between functors $ extbf{F}_n$ and $ extbf{H}_n$

Connected algebraic invariants with singular homology in a natural way

Linked the transformations to known results by Beilinson

Abstract

The category of topological spaces endowed with two marked points is equipped with two families $\mathbf F_n$ and $\mathbf H_n$ of functors to the category of abelian groups, indexed by a nonnegative integer $n$: namely, the functor $\mathbf F_n$ takes the object $(X,x,y)$ to the quotient of $\mathbb Z\pi_1(X,x,y)$ by an abelian subgroup associated with the $n+1$-st power of the augmentation ideal of the group algebra $\mathbb Z\pi_1(X,x)$, and the functor $\mathbf H_n$ takes the same object to the $n$-th singular homology group of $X^n$ relative to a subspace defined in terms of partial diagonals. We construct a family of natural transformations $\nu_n : \mathbf F_n\to \mathbf H_n$. We identify the natural transformation obtained by restricting $\nu_n$ to the subcategory of algebraic varieties with a natural equivalence due to Beilinson.