  We compute the rank of the fundamental group of an arbitrary connected
component of the space map(X, Y) for X and Y nilpotent CW complexes with X
finite. For the general component corresponding to a homotopy class f : X -->
Y, we give a formula directly computable from the Sullivan model for f. For the
component of the constant map, our formula expresses the rank in terms of
classical invariants of X and Y. Among other applications and calculations, we
obtain the following: Let G be a compact simple Lie group with maximal torus
T^n. Then the fundamental group of map(S^2, G/T^n; f) is a finite group if and
only if f: S^2 --> G/T^n is essential.
