Intersection of subgroups in free groups and homotopy groups

TL;DR

This paper explores the relationship between the intersection of three subgroups in free groups and the third homotopy group, generalizing previous results that linked two subgroups with the second homotopy group, and discusses applications to group homology.

Contribution

It introduces a natural homomorphism connecting the intersection of three subgroups in a free group to the third homotopy group, extending prior work on two subgroups.

Findings

The homomorphism is an isomorphism in certain cases.

The map relates subgroup intersections to $ ext{pi}_3$ of CW-complexes.

Applications to group homology are discussed.

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.