Questions about homotopic path

Question

Recently I am watching algebraic topology classes on youtube. When I watch the Lecture 24 which talked about homotopic path, I saw the professor show this picture and formulas: (link:https://www.youtube.com/watch?v=dkhuzT9zxxM at time 7:23)

the question picture

In this picture, the professor said the path $\alpha$ should be mapped to the interval t=[0,1] on x axis and the path $\beta$ should be mapped to the interval t=[0,1] which s=1. But I think the t axis is not equal to "time" right? Because H(0,s), as far as I am concerned, is the path on time 0, and the s is the actual interval we transformed to the path on our surface. So if I want to draw the path $\alpha$ and refer the t axis as "time", I should draw a vertical line not a horizontal line. Also in the rest of the video, he used the $\alpha$(t) for example: $\alpha$(t)=t to represent a path, so I am confused by these symbols.
So could anyone tell me whether I am right, or I what I understand is wrong?

Answer 1

The $s$ variable selects the path from the family of paths.

Thus,

• For $s=0$, $H(t,0)=\alpha(t)$, so $s=0$ yields the path $\alpha$.
• For $s=1$, $H(t,1)=\beta(t)$, so $s=1$ yields the path $\beta$.

For a fixed value of $s$, the $t$ variable is the parameter for the path with the given $s$-value.

When you say

• the path $\alpha$ should be mapped to the interval t=[0,1] on x axis

that should be corrected to

• for $s=0$, as $t$ varies from $0$ to $1$, the interval $[0,1]$ is mapped to the image of $\alpha$.

Similarly, when you say

• the path $\beta$ should be mapped to the interval t=[0,1] which s=1.

that should be corrected to

• for $s=1$, as $t$ varies from $0$ to $1$, the interval $[0,1]$ is mapped to the image of $\beta$.

Also, when you say

• H(0,s), as far as I am concerned, is the path on time 0

that's not a correct understanding of the meaning of $H(0,s)$.

For each fixed $s\in[0,1]$, $H(0,s)$ and $H(1,s)$ are the initial and final points, respectively, of the path with the given $s$-value.

Based on the definition of homotopy, $H(0,s)=\alpha(0)=\beta(0)$, and $H(1,s)=\alpha(1)=\beta(1)$, so $H(0,s)$ and $H(1,s)$ are constant (i.e., they don't vary as $s$ varies from $0$ to $1$).

If you choose to regard $t$ as "time", then it should be regarded as time for a fixed value of $s$ (i.e., for a particular path from the family of paths).

Thus, the presenter in the video is correct:

Each horizontal line segment of the unit square maps to the image of a path from the family of paths.