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 $K=K_1\cup K_2\cup K_3$ and $K_1\cap K_2\cap K_3$isthe1-skeleton$K^1$of$K$. We construct a natural homomorphism of ${\pi }_1(K)$-modules $${\pi }_3(K)\to \frac{R_1\cap R_2\cap R_3}{[R_1,R_2\cap R_3][R_2,R_3\cap R_1][R_3,R_1\cap R_2]},$$ where $R_i=ker\{{\pi }_1(K^1)\to {\pi }_1(K_i)\},i=1,2,3$ and the action of ${\pi }_1(K)=F\mathrm{/}{R}_1{R}_2{R}_3$ 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.