Two questions about homotopy of paths

Question

I'm a physics student and recently studying some homotopy theory. I have two questions based on understanding the definition of homotopy of paths.

1. For two paths $\alpha(s)$ and $\beta(s)$, in a topological space to be homotopic, is it necessary that their starting point and ending points should match i.e., is it required that $\alpha(0)=\beta(0)$ and $\alpha(1)=\beta(1)$? My physics book says it is so but I want to be sure.

2. If $\alpha(1)=\beta(0)$ and $\alpha(0)\neq \beta (1)$, can these paths be homotopic?

Answer 1

Homotopy of paths is almost always defined "rel endpoints", i.e., constraining the endpoints to remain fixed during the homotopy.

So for your second question, the answer is "no".

One small exception to the first rule is that when the paths are loops, we sometimes speak of a homotopy, though loops, from one loop to another where the basepoints are different. Such things are sometimes called "free homotopies" to distinguish them from the thing that everyone expects.

In general, it turns out that looking at homotopic maps from $A$ to $B$, where $A$ and $B$ are topological spaces, is interesting, but often more interesting is looking at homotopic maps from $(A, A')$ to $(B, B'),$ where $A' \subset A$ and $B' \subset B$. What this means is "maps from $A$ to $B$ that send the subset $A'$ into the subset $B'$.

For loops in a space $X$, based at some point $x_0$, we take $(A, A') = ([0, 1], \{0,1\})$ and $(B, B') = (X, \{x_0\})$. So we are looking at paths where both endpoints are constrained to always be at $x_0$.

For paths from $p$ to $q$, with a similar extension of notation, we're considering maps from $(A, A_1, A_2) = ([0, 1], \{0\}, \{1\})$, to $(B, B_1, B_2) = (X, \{p\}, \{q\})$.