The question is from Munkres: Consider the space $X$ obtained from a seven-sided polygonal region by means of the labelling scheme $abaaab^{-1}a^{-1}$. Show that the fundamental group of $X$ is the free product of two cyclic groups. The solution given says that the least normal subgroup $N$ is generated by $b^{-1}a^{-1}abaaa = a^3$. Can someone explain this?
Asked
Active
Viewed 327 times
3
-
If no one can answer this, can someone explain an alternative way to solve the problem? – ProbsNot Jan 14 '15 at 02:30
1 Answers
1
In Seifert–van Kampen theorem, you need to mod out the normal subgroup $N$ generated by $abaaab^{-1}a^{-1}$. But $(ab)aaa(ab)^{-1}=abaaab^{-1}a^{-1}$. Hence $N$ is also generated by $a^3$.
Hence $\pi(X)=\mathbb{Z_3} * \mathbb{Z}$.
WWK
- 1,370