Using Lemma 1.15, show that if a space $X$ is obtained from a path-connected subspace $A$ by attaching a cell $e^n$ with $n\geq 2$, then the inclusion $A\hookrightarrow X$ induces a surjection on $\pi_1$.
Proof. Note that $X = A\sqcup_{x\sim \phi(x)} e^n$ where $x\in S^{n-1}$ and $\phi:S^{n-1}\to A$ is an attaching map. My plan to use Lemma 1.15 is to using the fact that $X = A\cup e^n$ and $A,e^n$ are both open path connected subspace of $X$ and $\pi_1(e^n)=0$ for $n\geq 2$. But the problem is choosing a base point. $A\cap e^n$ is empty so there we can't choose base point contained both in $A$ and $e^n$. I need some manipulation to have two path connected open cover of $X$ whose intersection is also path connected. Following my plan, how can I choose such spaces?
Best Answer
It seems that you use Hatcher's edition from 2001. Other editions are not compatible with that.
Let $\xi \in e^n$, $A_1 = X \setminus \{\xi\}$, $A_2 = e^n$ and $x_0 \in e^n \setminus \{\xi\}$. Then $A$ is a strong deformation retract of $A_1$. I think you can complete the proof now.