On the second lower quotient of the fundamental group

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.

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