I'm just starting, and am familiar with some very basic homotopy theory ($\pi_1(S^1) \cong \mathbb{Z}$, Van-Kampen Theorem, the lifting lemma), but I can't seem to find a way to solve the following problem since it doesn't involve loops.
Consider two paths given by the left and right arcs of a circle around the origin,
$$p(t) = (\cos(t),\sin(t))\; \text{ and } \;q(t) = (\cos(t),-\sin(t))\;\text{ for }\;t\in (-\pi/2,\pi/2).$$
We have $\:p(-\pi/2) = q(\pi/2)\:$ and $\:p(\pi/2) = q(\pi/2),\:$ so they share the same endpoints but I have no idea how to show that these are not homotopic in $\;\mathbb{R}^2 \setminus {(0,0)}$.
Any help would be appreciated, particularly hints over complete answers. This is not homework.
You could take a $1$-form $$\omega= -\frac{y}{x^2 + y^2} dx +\frac{x}{x^2 + y^2} dy$$ and calculate $\int_p \omega$ and $\int_q \omega$. If $p$ and $q$ were homotopic with fixed ends then the integrals would be the same, because $\omega$ is a closed form ($d \omega =0$).