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Let $A: S^n \to S^n$ denote the map sending a point on a sphere to the exact opposite point. Let $A_*: H_n(S^n) \to H_n(S^n)$ denote the action of $A$ on the homology group. Then $A_*$ is an isomorphism from $\mathbb{Z}$ to $\mathbb{Z}$, but the generator might change.

For instance, on $S^1$, if $x_0$ is the base point on a clockwise loop and $x_1$ is a point a little clockwise of $x_0$, then their image under $A$ will have the same orientation, so $1$ will map to $1$ by $A_*$. On $S^2$, however, a clockwise loop on the top of the sphere will map to a counterclockwise loop on the bottom, so here $1$ maps to $-1$.

What can I say about higher dimensional spheres?

NoName
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  • It seems that what you want to compute is the degree (http://en.wikipedia.org/wiki/Degree_of_a_continuous_mapping#From_Sn_to_Sn) of the antipodal map. I suggest you to look at http://math.stackexchange.com/questions/452962/the-degree-of-antipodal-map – Dario Nov 14 '14 at 21:52
  • I agree that this is a duplicate of the question linked by Dario. – Kevin Carlson Nov 14 '14 at 22:12
  • BTW "clockwise loop on the top of the sphere" is null-homologous; indeed, $S^2$ is simply connected. –  Nov 15 '14 at 00:14

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