(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 $N$.
Rather than writing down the meaning 'describe above' and '$N$' in the statement, I think it's better to look p.49 of Hatcher's algebraic topology here.
The question is I can't understand the red line in the image. I understand why they're trying to choose such $\delta_\alpha$, but why such an element exists in $\pi_1(A\cap B,z_0)$? The map $\pi_1(A\cap B)\to \pi_1(A)$ is not surjective. Could you explain this?
Best Answer
To describe the desired loop $\delta_\alpha$ in detail, let's look at the parts of the path $\gamma_\alpha \phi_\alpha \bar\gamma_\alpha$, and describe corresponding new parts which will be concatenated together to define $\delta_\alpha$.
The path $\gamma_\alpha$ goes along the bottom of the strip labelled $S_\alpha$ in that picture, starting from its initial point $x_0$. Define a correponding new path $\gamma'_\alpha$ which instead goes along the top of the strip, starting from its initial point $z_0$.
The closed path $\phi_\alpha$ goes around the boundary circle of the disc $e^2_\alpha$, which is where that disc is attached to $X$, and the base point of $\phi_\alpha$ coincides with the terminal endpoint of $\gamma_\alpha$. Define a corresponding new path $\phi'_\alpha$ which instead goes around a circle in the disc $e^2_\alpha$ that is concentric to the outer circle, and such that the base point of $\phi'_\alpha$ meets the terminal endpoint of $\gamma'_\alpha$.
Thus the concatenation $\delta_\alpha = \gamma'_\alpha \, \phi'_\alpha \, \bar\gamma'_\alpha$ is a closed loop based at $z_0$, and $\delta_\alpha$ is entirely contained in $A \cap B$.
Furthermore, letting $h$ be "the line segment connecting $z_0$ to $x_0$ in the intersection of the $S_\alpha$'s", the concatenation $$\bar h \, \delta_\alpha \, h = \bar h \, \gamma'_\alpha \, \phi'_\alpha \, \bar\gamma'_\alpha \, h $$ is a closed loop in $A$ based at $x_0$ that is path homotopic to $\gamma_\alpha \, \phi_\alpha \, \bar\gamma_\alpha$. That path homotopy may be seen geometrically as moving through the strip $S_\alpha$ and through the annulus joining the outer circle $\phi_\alpha$ of $e^2_\alpha$ to the concentric circle $\phi'_\alpha$.
It follows that $\delta_\alpha$ is a loop "in $A \cap B$ based at $z_0$ representing the element of $\pi_1(A,z_0)$ corresponding to $[\gamma_\alpha\, \phi_\alpha\, \bar\gamma_\alpha]$ under the base-point change homomorphism $\beta_h$".