Change of fiber

If β is a path in B that starts at, say, b, then we have the homotopy ${\displaystyle h:p^{-1}(b)\times I\to I{\overset {\beta }{\to }}B}$ where the first map is a projection. Since p is a fibration, by the homotopy lifting property, h lifts to a homotopy ${\displaystyle g:p^{-1}(b)\times I\to E}$ with ${\displaystyle g_{0}:p^{-1}(b)\hookrightarrow E}$. We have:

${\displaystyle g_{1}:p^{-1}(b)\to p^{-1}(\beta (1))}$.

(There might be an ambiguity and so ${\displaystyle \beta \mapsto g_{1}}$ need not be well-defined.)

Let ${\displaystyle \operatorname {Pc} (B)}$ denote the set of path classes in B. We claim that the construction determines the map:

${\displaystyle \tau$ :\operatorname {Pc} (B)\to } the set of homotopy classes of maps.

Suppose β, β' are in the same path class; thus, there is a homotopy h from β to β'. Let

${\displaystyle K=I\times \{0,1\}\cup \{0\}\times I\subset I^{2}}$.

Drawing a picture, there is a homeomorphism ${\displaystyle I^{2}\to I^{2}}$ that restricts to a homeomorphism ${\displaystyle K\to I\times \{0\}}$. Let ${\displaystyle f:p^{-1}(b)\times K\to E}$ be such that ${\displaystyle f(x,s,0)=g(x,s)}$, ${\displaystyle f(x,s,1)=g'(x,s)}$ and ${\displaystyle f(x,0,t)=x}$.

Then, by the homotopy lifting property, we can lift the homotopy ${\displaystyle p^{-1}(b)\times I^{2}\to I^{2}{\overset {h}{\to }}B}$ to w such that w restricts to ${\displaystyle f}$. In particular, we have ${\displaystyle g_{1}\sim g_{1}'}$, establishing the claim.

It is clear from the construction that the map is a homomorphism: if ${\displaystyle \gamma (1)=\beta (0)}$,

${\displaystyle \tau ([c_{b}])=\operatorname {id} ,\,\tau ([\beta ]\cdot [\gamma ])=\tau ([\beta ])\circ \tau ([\gamma ])}$

where ${\displaystyle c_{b}}$ is the constant path at b. It follows that ${\displaystyle \tau ([\beta ])}$ has inverse. Hence, we can actually say:

${\displaystyle \tau$ :\operatorname {Pc} (B)\to } the set of homotopy classes of homotopy equivalences.

Also, we have: for each b in B,

${\displaystyle \tau$ :\pi _{1}(B,b)\to } { [ƒ] | homotopy equivalence ${\displaystyle f:p^{-1}(b)\to p^{-1}(b)}$ }

which is a group homomorphism (the right-hand side is clearly a group.) In other words, the fundamental group of B at b acts on the fiber over b, up to homotopy. This fact is a useful substitute for the absence of the structure group.

One consequence of the construction is the below:

• The fibers of p over a path-component is homotopy equivalent to each other.

• James F. Davis, Paul Kirk, Lecture Notes in Algebraic Topology
• May, J. A Concise Course in Algebraic Topology