  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 word raised to some power is a k-fold
nested product of commutators, in any group presented by generators and
relations.
