Question about loops(closed paths) in $\mathbb{S}^{1}$

Question

Let

$$\alpha \left(s \right) =\left( \cos{2\pi s},\sin{2\pi s}\right)$$ and $$\beta \left(s \right) =\ \left( \alpha \land\left( \alpha \land \overline{\alpha} \right)\right)\left( s \right)$$ with $s \in \left[0,1 \right]$ and $\overline{\alpha}=\alpha \left( 1-s \right)$ be two loops (closed paths) in $\mathbb{S}^1$.

With $\land$ defined as:

$$\left(\gamma_{1}\land\gamma_{2}\right)\left(s\right)=\begin{cases} \gamma_{1}\left(2s\right) & s\in\left[0,\frac{1}{2}\right]\\ \gamma_{2}\left(2s-1\right) & s\in\left[\frac{1}{2},1\right] \end{cases}$$

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These are the things that I want to find:

1. An explicit parametrization of $\beta$.
2. An homotopy lifting of $\beta$, $\beta ':I \rightarrow \mathbb{R}$ so that $\beta '\left( 0 \right)=1$.
3. Homotopy $F \left(s,t \right)$ between $\beta$ and $\alpha$ relative to the point $\left( 0,1\right)$, in other words, so that $F \left(0,t \right)=F \left(1,t \right)= \left(1,0 \right)$ $\forall t$.

Any ideas? I will edit this post with my advances so I with your help can develop an answer

Answer 1

Here are some ideas for the respective exercises:

1. Write out the definition.
2. A lifting to where?
3. The loop $\beta$ first traces the loop $\alpha$ twice, and then traces back $\alpha$ in reverse. To construct a homotopy, contract the second part to a point. (By the second part, I mean the part tracing $\alpha$ once and then tracing $\alpha$ back in reverse). You can achieve this by a homotopy of the source spaces, i.e. a homotopy $[0,1]\ \longrightarrow\ [0,1]$. Here's a picture of this homotopy: