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 $H_{n}(S^{n})$?
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The question does not make sense because $\sigma_1$ and $\sigma_2$ are not chains, not with respect to singular homology or with respect to any other kind of homology I can think of that you might mean. – Lee Mosher Aug 27 '14 at 13:24