First, let me state the proposition in Hatcher's textbook
(a) If $Y$ is obtained from $X$ by attaching 2-cells as described above, then the inclusion $X\hookrightarrow Y$ induces a surjection $\pi_1(X,x_0)\to\pi_1(Y,x_0)$ whose kernel is $N$. Thus $\pi_1(Y)\simeq\pi_1(X)/N$
Note that $N$ is a normal subgroup of $\pi_1(X,x_0)$ generated by all the loops $\gamma_\alpha\varphi_\alpha\gamma_\alpha^{-1}$. I'm wondering the orientation $\varphi_\alpha$ goes. Let me just write an example:
If I want to attach single 2-cell to below 1-skeleton…
…like this
Then there are several ways of attaching 2-cell. For example, attaching 2-cell along $aba^{-1}b^{-1}cbc^{-1}$ or $aba^{-1}b^{-1}cb^{-1}c^{-1}$ or $cbc^{-1}aba^{-1}b^{-1}$…etc. Is this attaching orientation(?) matters? I mean even attached orientations are different but still, those spaces are homeomorphic? Or at least the fundamental groups are always isomorphic?
Best Answer
I think it's not true in general that the orientation of attaching cells is irrelevant to the resulting attached space.
Consider the following example:
The following is 1-skeleton
I'm going to attach 2-cell as the following image.
There're essentially two ways to attach 2-cell. One is along $abab^{-1}cac^{-1}$ and the other is $aba^{-1}b^{-1}ca^{-1}c^{-1}$. Let $X_1$ be $X^2$ whose 2-cell is attached by the former way and $X_2$ be $X^2$ whose 2-cell is attached by the latter way.
Then, $\pi_1(X_1)=\langle a,b,c|abab^{-1}cac^{-1}\rangle$ and $\pi_1(X_2)= \langle a,b,c|aba^{-1}b^{-1}ca^{-1}c^{-1}\rangle$. Then $\pi_1(X_1)^{\text{ab}}=\Bbb Z_3\oplus\Bbb Z\oplus\Bbb Z$ and $\pi_1(X_2)^{\text{ab}} = \Bbb Z\oplus\Bbb Z$ so that $X_1$ and $X_2$ are not homeomorphic (even in the level of fundamental group).