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 π3. This generalizes a result of Gutierrez-Ratcliffe who relate the intersection of two subgroups with the computation of π2. Let K be a twodimensional CW-complex with subcomplexes K1,K2,K3 such that K = K1 ∪ K2 ∪ K3 and K1 ∩ K2 ∩ K3 is the 1-skeleton K of K. We construct a natural homomorphism of π1(K)modules π3(K) → R1 ∩R2 ∩R3 [R1, R2 ∩R3][R2, R3 ∩R1][R3, R1 ∩R2] , where Ri = ker{π1(K1) → π1(Ki)}, i = 1, 2, 3 and the action of π1(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.