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Fibrations

A locally trivial fibration is a surjective map of topological spaces $f\:E\to B$ such that for every $b\in B$ there exists an neighbourhood $U_b$ of $b$ in $B$ such that $f^{-1}(U_b)\isom U_b\times F$, where $F$ is a fiber.


\begin{example} The M\uml {o}bius band is a fibration over $S^1$. It is not a trivial fibration because it is not a product. \end{example}

There is a fibration

\begin{displaymath} G\to \EG\to \BG \end{displaymath}

where $\EG$ is a contractible space. For example if $G=\bZ$, then this fibration is homotopy equivalent to

\begin{displaymath} \bZ\to\bR\to S^1 \end{displaymath}

Figure: $\bZ\hookrightarrow \bR\to S^1$
./Chapter1/R_to_S1.eps
But $\B\bZ$ is not a space with one 0-cell and one 1-cell. The 0-cells are in bijection with $\bZ$, and 1-cells are in bijection with pairs of distinct integers.


\begin{example} % latex2html id marker 277The \textbf{Hopf fibration} it is a ... ...S^1\times D^2\cup_{S^1\times S^1} D^2\times S^1$} \end{figure}\par \end{example}
If $X$ and $Y$ are pointed spaces, then we can perform the join construction $X*Y$.

\begin{displaymath} X*Y:=X\times I\times Y/ \sim, \end{displaymath}


\begin{displaymath} (x, 0, *)\sim (x',0, *) \end{displaymath}


\begin{displaymath} (*, 1, y)\sim (*, 1, y) \end{displaymath}

For example $S^1 * S^1=S^3$.
\begin{exer} Show that $\Dl^p* \Dl^q\isom \Dl^{p+q+1}$. \end{exer}


Pawel Witkowski 2006-11-07