Natural transformations relating homotopy and singular homology functors

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 $\mathbb Zπ_1(X,x,y)$ by an abelian subgroup associated with the $n+1$-st power of the augmentation ideal of the group algebra $\mathbb Zπ_1(X,x)$, 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 $ν_n : \mathbf F_n\to \mathbf H_n$. We identify the natural transformation obtained by restricting $ν_n$ to the subcategory of algebraic varieties with a natural equivalence due to Beilinson.

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