joinJoin32003-02-06 22:21:392004-09-13 17:52:07Definitionjoin% used for TeXing text within eps files
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\newcommand{\codim}{\operatorname{codim}}Given two topological spaces $X$ and $Y$, their {\em join,} denoted by $X\star Y,$ is defined to be the quotient space
\[
X\star Y := X\cross[0,1]\cross Y/\sim,
\]
where the equivalence relation $\sim$ is generated by
\begin{eqnarray*}
(x,0,y_1)& \sim (x,0,y_2) &\text{for any}\, x\in X,\, y_1,y_2\in Y,\, \text{and}\\
(x_1,1,y)& \sim (x_2,1,y) &\text{for any}\, y\in Y,\, x_1,x_2\in X.
\end{eqnarray*}
Intuitively, $X\star Y$ is formed by taking the disjoint union of the two spaces and attaching a line segment joining every point in $X$ to every point in $Y.$
Some examples:
\begin{itemize}
\item The join of a space $X$ with a one-point space is called the \emph{cone} of $X$.
\item The join of the spheres $S^n$ and $S^m$ is the sphere $S^{n+m+1}$.
\end{itemize}