Intersection of subgroups in free groups and homotopy groups

Hans-Joachim Baues; Roman Mikhailov

Intersection of subgroups in free groups and homotopy groups

Group Theory 2010-09-01 v1 Algebraic Topology

Authors: Hans-Joachim Baues , Roman Mikhailov

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.

Keywords

homotopy groups intersection theory algebraic topology

Cite

@article{arxiv.0804.1999,
  title  = {Intersection of subgroups in free groups and homotopy groups},
  author = {Hans-Joachim Baues and Roman Mikhailov},
  journal= {arXiv preprint arXiv:0804.1999},
  year   = {2010}
}

Intersection of subgroups in free groups and homotopy groups

Abstract

Keywords

Cite

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