Natural transformations relating homotopy and singular homology functors

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.

in French language