homotopy equivalenceHomotopyEquivalence2002-01-23 14:14:462003-07-10 02:36:45Definitionhomotopy equivalenthomotopically equivalenthomotopy typestrong homotopy equivalence% this is the default PlanetMath preamble. as your knowledge
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% define commands here{\bf Definition} Suppose that $X$ and $Y$ are topological spaces and
$f: X \to Y$ is a continuous map.
If there exists a
continuous map $g:Y \to X$ such that $f\circ g \simeq id_{Y}$
(i.e. $f\circ g$ is \PMlinkid{homotopic}{1584} to the identity
mapping on $Y$),
and $g \circ f \simeq id_{X}$, then
$f$ is a \emph{homotopy equivalence}.
This homotopy equivalence is sometimes called
\emph{strong homotopy equivalence} to distinguish it from
weak homotopy equivalence.
If there exist a homotopy equivalence between the topological
spaces $X$ and $Y$, we say that $X$ and $Y$ are
\emph{homotopy equivalent}, or that
$X$ and $Y$ are of the same \emph{homotopy type}.
We then write $X\simeq Y$.
\subsubsection{Properties}
\begin{enumerate}
\item Any homeomorphism $f:X\to Y$ is obviously a homotopy equivalence with
$g=f^{-1}$.
\item For topological spaces, homotopy equivalence is an
equivalence relation.
\item A topological space $X$ is (by definition) contractible,
if $X$ is homotopy equivalent to a point, i.e., $X\simeq \{x_0\}$.
\end{enumerate}
\begin{thebibliography}{9}
\bibitem{hatcher_at} A. Hatcher, \emph{Algebraic Topology}, Cambridge University Press, 2002. Also available
\PMlinkexternal{online}{http://www.math.cornell.edu/~hatcher/AT/ATpage.html}.
\end{thebibliography}