On the second lower quotient of the fundamental group

TL;DR

This paper investigates the second lower quotient of the fundamental group of a topological space, relating it to homology and cohomology, and provides an example involving Fano surfaces to illustrate the theory.

Contribution

It expresses the second quotient of the lower central series of the fundamental group in terms of homology and cohomology, offering a new perspective and recovering known results.

Findings

Derived an expression for D/(D,G) in terms of homology and cohomology.

Reproduced the known isomorphism D/(D,G) = Z/2 for Fano surfaces.

Established a connection between fundamental group quotients and topological invariants.

Abstract

Let X be a reasonable topological space, G its fundamental group, and D = (G,G). We express the second quotient D/(D,G) of the lower central series of G in terms of the homology and cohomology of X . As an example, we recover the isomorphism D/(D,G) = Z/2 (due to Collino) when X is the Fano surface parametrizing lines in a cubic threefold.