Let $S$ be a closed, orientable surface and $\gamma$ a simple closed curve on $S$. Show that $S$ retracts onto $\gamma$ if and only if $\gamma$ represents a non-trivial element in homology.
Clearly, if $[\gamma]=0$ in $H_1(S)$, the composition $H_1(\gamma) \rightarrow H_1(S) \rightarrow H_1(\gamma)$ can not be the identity. If the class of $\gamma$ is non-zero, how can we construct such retraction?
use "change of coordinates principle". this is an easy consequence of the classification of surfaces with a boundary, it principle states:
If $\alpha$ and $\beta$ are any two nonseparating simple closed curves in a surface $S$, then there is a homeomorphism $\phi : S \to S$ with $\phi(\alpha)=\beta$.
therefore, you may asume that given curve is a meridian. then you can construct "pinch" map from $S$ to a torus, such that your curve maps to the meridian of this torus, and then retract torus to the circle.
