boundary in homology group

Question

f is a reflection on a sphere $S^{n}$, $\sigma_{1}$ is a diffeomorphism from $D^{n}\subset \mathbb{R}^{n}$ to one of the two caps of the sphere, separated by the plane of the reflection and $\sigma_{2}=f(\sigma_{1})$. How can I show that $\sigma_{1}- \sigma_{2}$ is not a boundary in $H_{n}(S^{n})$?