Fundamental group obtained by attaching a n-cell with n ≥ 2 I am having trouble with Hatcher's Algebraic Topology P39, Problem 18:
Show that if a space $X$ is obtained from a path-connected subspace $A$ by attaching a cell $e^n$ with $n ≥ 2$, then the inclusion $A \rightarrow X$ induces a surjection on $π_1$.
The given hint is to follow the proof of Proposition 1.14, which states that $\pi_1(S^n) = 0$ if $n \geq 2$. The proof of 1.14 roughly goes: any given loop can be moved to avoid any specific point. Then, if a loop is disjoint from a point $x$, we can deformation retract $S^n - x$ onto a single point, thus killing the loop.
Applying this method to Problem 18 seems easy: I can show that if a loop is disjoint from any point $y \in e^n \backslash \delta e^n$, then I can deformation retract $e^n \backslash y$ onto $\delta e^n \subseteq A$. And, moving the loop off of $y$ is doable. Thus, any loop in $X$ becomes a loop in $A$. However, this doesn't get me to the result; loops in $X$ can still travel around $\delta e^n$, which adds connections between points in $A$ that didn't exist before.
To make this visual, I have a "counterexample" to the problem. 18a says "The wedge sum $S^1 ∨ S^2$ has fundamental group $\mathbb{Z}$." However, using the result from the problem, I can construct:
Let $A$ consist of two points $a$ and $b$, and a line between them. Then, attach $D^2$ by sending $e^{i \pi k} \rightarrow a$ for $0 \leq k < 1$, and $e^{i \pi j} \rightarrow b$ for $1 \leq j < 2$. This turns $D^2$ into a sphere, with $a$ and $b$ being two points infinitesimally close to each other on its surface, so our construction is homotopy equivalent to $S^1 ∨ S^2$. However, $\pi_1(A) = 0$, while $\pi_1(X) = \mathbb{Z}$.
So my question(s) is:


*

*How to actually prove Problem 18?

*What is wrong with my counterexample?

*Are $a$ and $b$ actually sent to infinitesimally close points?

 A: Reading the comments it seems your second and third questions have been answered. I address the first question.
The hint suggests using Lemma 1.15, which reads:

If a space $A$ is the union of a collection of path-connected open sets $A_\alpha$, each containing the basepoint $x_0\in A$ and if each intersection $A_\alpha\cap A_\beta$ is path-connected, then every loop in $A$ at $x_0$ is homotopic to a product of loops each of which is contained in a single $A_\alpha$.

Letting $A_\alpha=e^n-x$ where $x\in e^n-\partial e^n$, and $A_\beta=X$, we see that $A_\alpha\cap A_\beta=\partial e^n$ which is path connected and $A=A_\alpha\cup A_\beta$: the lemma will apply. Every loop in $A$ at $x_0$ is some product of loops $p_1\cdots p_n$, which is homotopic to a product of loops in $X$; since any $p_i\in\pi_1(A_\alpha)$ is nullhomotopic ($A_\alpha$ deformation retracts onto $S^k$, $k\ge 2$, then use Proposition 1.14). This is exactly what was required.

A similar result
After Section 1.2, we can use the van Kampen theorem to get a similar result: a CW complex $X$ and its two-skeleton $X^2$ have isomporphic fundamental groups.
Proof. Since $X^n$ is the quotient space of the disjoint union $X^{n-1}\coprod_\alpha D_\alpha^n$ with identifications $x\sim\phi_\alpha(x)$ for $x\in\partial D_\alpha^n$, it suffices to show that $\pi_1(X^{n-1}\coprod D^n)\cong\pi_1(X^{n-1})$ for $n\ge 3$.
Let $x\in D^n-\partial D^n$, and $A_\alpha=D^n-x$; and $A_\beta=X^{n-1}$. Then $X^{n-1}\coprod D^n=A_\alpha\cup A_\beta$, and $A_\alpha\cap A_\beta=\partial D^n$ which is isomorphic to $S^{n-1}$ which is path connected. By the van Kampen theorem, $$\pi_1(X^n)\cong \pi_1(A_\alpha)\ast\pi_1(A_\beta)\cong\pi_1(S^{n-1})\ast\pi_1(X^{n-1})\cong \pi_1(X^2),$$
using an induction argument, and noting that $\pi_1(S^{n-1})\cong 0$ for $n\ge 2$. In particular, since $\pi_1(X^n)\cong\pi_1(X^2)$ for all $n$, $\pi_1(X)\cong\pi_1(X^2)$.
