If I am not wrong the homotopy type of curve in the punctured plane is dictated by winding number around the puncture.
This is not the case for the doubly punctured plane (say with puncture at the points $p,q$), since the loop $γ_1$ forming loops first at $p$ then at $q$ and then at $p$ again is not homotpic at the loop $γ_2$ which forms first 2 loops around $p$ and then one around $q$.
I have shown $H^1(\mathbb{R^2}-{p})=1$ and that $H^1(\mathbb{R^2}-{p,q})=2$. I was wondering if this result is connected to that one above and/or if there is any direct way of proving that.Maybe one way to prove this would be the DeRham Theorem but since I don't know any homology theory I can't make any use of it.
So, to make a concrete question. Is the fact about the winding number dictating the homotopies in $\mathbb{R^2}-\{p\}$ and not $\mathbb{R^2}-\{p,q\}$ related with the dimension of the dimension of $H^1(\mathbb{R^2}-{p})$ and $H^1(\mathbb{R^2}-{p,q})$? I would like a prefer an intuitive explanation as to why or why not to a proof although that would also be very appreciated.