# Attaching a $1-$cell to a space induces an injection between fundamental groups

Consider a path-connected topological space $$X$$, to which we attach a $$1-$$cell via the map $$\phi:S^0 \to X$$, where $$S^0 = \{-1,1 \}$$. The space we obtain is $$Y = (X \sqcup [-1,1]) / \{-1 \sim \phi(-1) \text{ and } 1 \sim \phi(1) \}.$$

How can we prove that the inclusion $$i:X \hookrightarrow{} Y$$ induces an injective homomorphism $$i_*:\pi_1(X,p) \to \pi_1(Y,p)$$ for every $$p \in X$$?

Intuitively, I see that the attaching a $$1-$$cell to the space $$X$$ is similar to $$X \lor S^1$$ (in the case that we can deformation retract the path between $$\phi(1)$$ to $$\phi(-1)$$ in $$X$$ to a point), so the fundamental group of $$Y$$ is $$\pi_1(X) * \mathbb{Z}$$. But what if we cannot deformation retract this path to a point? Also, how do we specifically prove that the inclusion induces an injective homomorphism (we know that it always is a homomorphism); computing the fundamental group of $$Y$$ using the theorem of Seifert-van Kampen doesn't seem to suffice.

As $$X$$ is path connected, we can define a continuous map $$f\colon Y\to X$$ that is the identity on $$X$$ and maps the glued cell to a path in $$X$$ between the glue points. Now consider a loop that is in the kernel of $$i_*$$ and apply $$f$$ to the corresponding homotopy.
• So if $\gamma$ is a loop in $X$ at $p$ with $[i \circ \gamma] = i_*([\gamma])$ being the constant loop in $Y$ at $p$, we want to prove that $\gamma$ is the constant loop in $X$ in $p$. Applying $f$ to the homotopy between $i \circ \gamma$ and the constant loop gives a homotopy between $f\circ i \circ \gamma$ and the constant loop at $f(p)$ (both of them being loops in $X$), but how do we proceed from here? – Otp Mar 8 at 10:20
• @Otp We have $f \circ i = id$, thus $f_* \circ i_* = id$. This implies that $i_*$ is injective. – Paul Frost Mar 8 at 10:52