antipodalAntipodal2003-09-13 03:51:172006-01-14 15:02:16Definitionantipodal pointsantipodal map% this is the default PlanetMath preamble. as your knowledge
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\makeatother{\bf Definition}
Suppose $x$ and $y$ are points on the
\PMlinkname{$n$-sphere}{Sphere} $S^n$.
If $x=-y$ then $x$ and
$y$ are called {\bf antipodal points}. The
{\bf antipodal map} is the map $A: S^n\to S^n$ defined as
$A(x)=-x$.
\textbf{Properties}
\begin{enumerate}
\item
The antipodal map$A:S^n\to S^n$ is
\PMlinkname{homotopic}{HomotopyOfMaps}
to the identity map if $n$ is odd \cite{guillemin}.
\item The \PMlinkname{degree}{DegreeMapOfSpheres} of the
antipodal map is $(-1)^{n+1}$.
\end{enumerate}
\begin{thebibliography}{9}
\bibitem{guillemin} V. Guillemin, A. Pollack,
\emph{Differential topology}, Prentice-Hall Inc., 1974.
\end{thebibliography}