Consider the plane with a point removed $M=\mathbb{R}^2\setminus\{0\}$. Let $p_1,p_2\in M$ be two points. I want to understand how to classify the homotopy classes of paths joining $p_1$ and $p_2$.
Let's take for example $p_1 = (a,0)$ and $p_2 = (0,a)$ with both $a>0$. One possibility of path from $p_1$ to $p_2$ is the circular arc $\gamma_1:[0,1]\to M$ given by $$\gamma_1(s) = \left(a\cos\frac{\pi s}{2},a\sin\frac{\pi s}{2}\right).$$
On the other hand we could also consider $$\gamma_2(s) = \left(a\cos\left(-\frac{3\pi s}{2}\right),a\sin\left(-\frac{3\pi s}{2}\right)\right).$$
It is intuitively clear they can't be homotopic. If a homotopy $F:[0,1]\times [0,1]\to M$ existed, it seems intuitively clear to me that because of continuity there had to be some $t_\ast\in (0,1)$ such that $F(t_\ast,s)=(0,0)$ for some $s$ which would not be possible in $M$.
While intuitively clear, I don't know how to come with up a proof right now. Also, I'm finding it challenging to have intuition about paths that go around the origin more than once.
So, can someone give a hint on how to turn this intuitive picture into a rigorous proof and how to deal with the case in which the path goes around the origin multiple times? Also, how can I be sure that I have enumerated all possible classes? I would prefer a hint instead of a complete answer.
Also, some guidance in generalizing this to generic points $p_1$ and $p_2$ would be nice.
Fix basepoints in $X=\mathbb{R}\backslash\{0\}$, and fix a path $T:[0,1]\to X$ between basepoints. Now given two paths $f,g:[0,1]\to X$ it is not hard to see that these are homotopic if and only if $f*T^{-1}$ and $g*T^{-1}$ are. Here I'm talking about path composition and $T^{-1}$ is the reversed $T$. Note that $f*T^{-1}$ and $g*T^{-1}$ are loops at single basepoint.
In other words we can reduce this problem to studying loops. And this can be done for example via winding number. It is a precise invariant that tell us how many times a loop goes around zero. It turns out that two loops are homotopic if and only if they have the same winding number. A more general variant of this is known as Hopf theorem.
If you apply this to your case and take $T=\gamma_1$ then $\gamma_1*T^{-1}$ has winding number $0$: it goes one way forward and back, never around zero. While $\gamma_2*T^{-1}$ has winding number $\pm 1$ because it goes around zero one time. The sign tells us in which direction precisely, but I don't remember from top of my head whether plus denotes counterclockwise or the other way around.
Think about $\gamma_k(t)=exp(k 2\pi i t)$ for an integer $k$. Its like spinning really fast in order to make $k$ wrappings around origin in $[0,1]$ time interval. And in fact the winding number of $\gamma_k$ is $k$ and so any loop is homotopic to some $\gamma_k$. While $\gamma_k$ is not homotopic to $\gamma_s$ for $k\neq s$.
And to talk about going around origin for arbitrary paths (not necessarily loops), my advice is to reduce the problem to loops as I did at the beginning.