Let $a,b,c \in X$ some path-connected space. Show a path from $a$ to $b$ is homotopic to path from $a$ to $b$ passing through $c$ in path-connected space.
My attempt:
Let $\gamma_0$ be a path from $a$ to $b$ let $\gamma_1$ be a path from $a$ to $c$ and let $\gamma_2$ be a path from $c$ to $b$. Then let $\gamma$ be the product of paths $\gamma_1,\gamma_2$. I.e.,
$$\gamma(t):=\begin{cases}
\gamma_1(2t) & t \in [0,\frac{1}{2}] \\
\gamma_2(2t-1)& t \in [\frac{1}{2},1]
\end{cases}$$
Then can I define a homotopy from $\gamma_0$ to $\gamma$ via
$$H(s,t):= \begin{cases}
\gamma_0(2s,t) & s \in [0,\frac{1}{2}]\\
\gamma(2s-1,t) & s \in [\frac{1}{2},1]
\end{cases}
$$
or do I need to break up the intervals more? Thanks in advance for any tips or hints, I'm fairly new to alg top. Also any text recommendations for alg top for dummies type? We're using the last half of Munkres together with the classical Hatcher.
Best Answer
As $X$ is path-connected, any two points can be adjoined via a path. Let $a,b \in X$. Define $\gamma(t)$ as $$\gamma:[0,1] \to X$$ such that $$\gamma(0)=a, \gamma(1)=b.$$ Let $c \in X$ and define $$\sigma:[0,1] \to X$$ such that $$\sigma(0)=a, \sigma(1)=c.$$ Then if we define $$\sigma^{-1}:=\sigma(1-t)$$ then $\sigma^{-1}$ goes from $c$ back to $a$ thus $\sigma \star \sigma^{-1}$ is the constant path, based at $a$, denoted $e_a(t).$ Then $$(\sigma \star \sigma^{-1} \star \gamma)(t) \simeq e_s(t) \gamma(t).$$ is a path from $a$ to $b$ passing through $c$. Thus $$(\sigma \star \sigma^{-1} \star \gamma)(t) \simeq \gamma(t).$$ Thus the two are homotopic.