Klein bottle in $3$ dimension named $Y$ is given here (the circle of self-intersection is deleted).
Its CW complex structure is given at right.
$\pi_1(Y)$ is generated by $a, b, c, d$ with $cbc^{-1}d=1, aba^{-1}b^{-1}d^{-1}=1$.
It can be reduced to $aba^{-1}b^{-1}cbc^{-1}=1$, so $\pi_1(Y)=\langle a,b,c \mid aba^{-1}b^{-1}cbc^{-1}=1 \rangle$.
Hatcher's book Algebraic Topology in page 53-54 says $\pi_1(Y)$ also has presentation $\langle a,b,c \mid aba^{-1}b^{-1}cb^\color{red}{-1}c^{-1}=1 \rangle$ (this gives isomorphic group as above).
My question:
$1$. How can we find a CW complex structure s.t. $\pi_1(Y)$ has presentation $\langle a,b,c \mid aba^{-1}b^{-1}cb^\color{red}{-1}c^{-1}=1 \rangle$?
$2$. How can we show $\langle a,b,c \mid aba^{-1}b^{-1}cbc^{-1}=1 \rangle\cong \langle a,b,c \mid aba^{-1}b^{-1}cb^\color{red}{-1}c^{-1}=1 \rangle$ by giving explicit isomorphism?
Thanks for your times and effort.
Best Answer
I didn't find the desired CW complex structure, but I found the isomorphism between groups.
$\pi_1(Y)=\langle a, b, c, d \mid cbc^{-1}d=1, aba^{-1}b^{-1}d^{-1}=1\rangle=\langle a,b,c \mid aba^{-1}b^{-1}cbc^{-1}=1 \rangle$, deoted by $G$.
Replace $c$ by $ad$, then $a,b,d$ are generators of $G$ and
$aba^{-1}b^{-1}cbc^{-1}=1$ becomes $a^{-1}bab^{-1}db^{-1}d^{-1}=1$.
Replace $d$ by $c'$ and $a^{-1}$ by $a'$, then $a',b,c'$ are generators of $G$ and
$a^{-1}bab^{-1}db^{-1}d^{-1}=1$ becomes $a'ba'^{-1}b^{-1}c'b^{-1}c'^{-1}=1$.
Therefore $\langle a,b,c \mid aba^{-1}b^{-1}cbc^{-1}=1 \rangle\cong \langle a,b,c \mid aba^{-1}b^{-1}cb^{-1}c^{-1}=1 \rangle$.