Infinitesimal calculations in fundamental groups

We show that Hopf invariants, defined by evaluation in Harrison cohomology of the commutative cochains of a space, calculate the logarithm map from a fundamental group to its Malcev Lie algebra. They thus present the zeroth Harrison cohomology as a universal dual object to the Malcev Lie algebra. This structural theorem supports explicit calculations in algebraic topology, geometric topology, and combinatorial group theory. In particular, we give the first algorithm to determine whether a power of a word is a k-fold nested commutator while encoding commutator structure in any group presented by generators and relations.