weak homotopy equivalenceWeakHomotopyEquivalence2003-02-07 02:07:092008-10-06 22:42:08Definitionweakly homotopy equivalentweakly equivalent% used for TeXing text within eps files
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A continuous map $\funcdef{f}{X}{Y}$ between path-connected based
topological spaces is said to be a {\em weak homotopy equivalence\/} if for each $k\ge 1$ it induces an isomorphism $\funcdef{f_*}{\pi_k(X)}{\pi_k(Y)}$ between the
$k$th homotopy groups. $X$ and $Y$ are then said to be {\em weakly
homotopy equivalent.}
\begin{rmk}
It is {\em not\/} enough for $\pi_k(X)$ to be isomorphic to
$\pi_k(Y)$ for all $k.$ The definition requires these isomorphisms
to be induced by a space-level map $f.$
\end{rmk}
\begin{rmk}
More generally, two spaces $X$ and $Y$ are defined to be weakly homotopy equivalent if there is a sequence of spaces and maps
$$ X \to X_1 \from X_2 \to X_3 \from \cdots \to X_n \from Y$$ in which each map is a weak homotopy equivalence.
\end{rmk}