Intersection of subgroups in free groups and homotopy groups

Abstract

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 $K_1\cap K_2\cap K_3$ is the 1-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 $\pi_1(K)=F/R_1R_2R_3$ 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.