A example of infinite homotopy class of curve from p to q

Question

For a differentiable manifold $M$ and $ p,q \in M$ ,let $$ C_{pq}=\{\text{all curves from p to q}\} \\ H_{pq}=C_{pq}/\sim $$ $\sim$ is homotopy equivalence. Whether there is a differentiable manifold $M$ (finite dimensional) st there are two points $p,q\in M$ st $H_{pq}$ is a infinite set ?

Answer 1

Let $M=\Bbb{C} \setminus \{0\}$, and let $p=q=1$. Then $M$ is a 2-dimensional manifold but for each integer $n$, I can find a loop $\gamma_n$ based at $1$ that winds $n$ times counter-clockwise around $0$. Then if $n \ne m$, there is no hope of finding a homotopy from $\gamma_n$ to $\gamma_m$.

This can be proved with basic complex analysis, even if you know no algebraic topology. Thanks Mariano for singling out a "smallest" example.