Rank of the fundamental group of any component of a function space

Abstract

We compute the rank of the fundamental group of any connected component of the space map(X, Y) for X and Y connected, nilpotent CW complexes of finite type with X finite. For the component corresponding to a general 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 retrieves a known expression for the rank in terms of classical invariants of X and Y. When both X and Y are rationally elliptic spaces with positive Euler characteristic, we use our formula to determine the rank of the fundamental group of any component of map(X, Y) explicitly in terms of the homomorphism induced by f on rational cohomology.