In topology:
Regarding the n-sphere as the unit sphere inside Cartesian space, $S^n \,\simeq\, S(\mathbb{R}^{n+1})$, the antipode of any point $p \,\in\, S^n \hookrightarrow \mathbb{R}^{n=1}$ is the point $-p \,\in\, S^n \hookrightarrow \mathbb{R}^{n+1}$ obtained by sending the coordinates of $p$ to their negatives.
The quotient space of the n-sphere by the $\mathbb{Z}/2$-action given by switching antipodes is the real projective space $\mathbb{R}P^{n-1}$.
See also equator, hemisphere, meridian.
In algebra:
The antipode of a Hopf algebra is the formal dual of the operation of passing to inverse elements in a group.
If the Hopf algebra is the Pontrjagin ring-structure on the homology of loop spaces, then this antipode-operation corresponds to reversal of orientation of loops.
See also
Last revised on January 5, 2023 at 18:57:33. See the history of this page for a list of all contributions to it.