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?