nLab antipode

Redirected from "antipodes".

Literature

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.