Intersection of subgroups in free groups and homotopy groups

We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group pi(3). This generalizes a result of Gutierrez-Ratcliffe who relate the intersection of two subgroups with the computation of pi(2). Let K be a two-dimensional CW-complex with subcomplexes K-1, K-2, K-3 such that K = K-1 boolean OR K-2 boolean OR K-3 and K-1 boolean AND K-2 boolean AND K-3 is the 1-skeleton K-1 of K. We construct a natural homomorphism of pi(1)(K)-modules [GRAPHICS] where R-i = ker{pi(1)(K-1) -> pi(1)(K-i)}, i = 1, 2,3 and the action of pi(1)( K) = F/R1R2R3 on the right-hand abelian group is defined via conjugation in F. In certain cases, the defined map is an isomorphism. Finally, we discuss certain applications of the above map to group homology.