Change of fiber

In algebraic topology, given a fibration p:E→B, the change of fiber is a map between the fibers induced by paths in B. Since a covering is a fibration, the construction generalizes the corresponding facts in the theory of covering spaces.

Definition

If β is a path in B that starts at, say, b, then we have the homotopy $h:p^{-1}(b)\times I\to I\stackrel{\beta }{\to }B$ where the first map is a projection. Since p is a fibration, by the homotopy lifting property, h lifts to a homotopy $g:p^{-1}(b)\times I\to E$ with $g_0:p^{-1}(b)\hookrightarrow E$. We have:

$g_1:p^{-1}(b)\to p^{-1}(\beta (1))$.

(There might be an ambiguity and so $\beta \mapsto g_1$ need not be well-defined.) Let $\mathrm{Pc}(B)$ denote the set of path classes in B. We claim that the construction determines the map:

$\tau :\mathrm{Pc}(B)\to$ 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

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

Drawing a picture, there is a homeomorphism $I^2\to I^2$ that restricts to a homeomorphism $K\to I\times \{0\}$. Let $f:p^{-1}(b)\times K\to E$ be such that $f(x,s,0)=g(x,s)$, $f(x,s,1)=g^{\prime }(x,s)$ and $f(x,0,t)=x$. Then, by the homotopy lifting property, we can lift the homotopy $p^{-1}(b)\times I^2\to I^2\stackrel{h}{\to }B$ to w such that w restricts to $f$. In particular, we have $g_1\sim {g_1}^{\prime }$, establishing the claim. It is clear from the construction that the map is a homomorphism: if $\gamma (1)=\beta (0)$,

$\tau ([c_b])=\mathrm{id},\tau ([\beta ]\cdot [\gamma ])=\tau ([\beta ])\circ \tau ([\gamma ])$

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

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

Also, we have: for each b in B,

$\tau :{\pi }_1(B,b)\to$ { [ƒ] | homotopy equivalence $f:p^{-1}(b)\to p^{-1}(b)$ }

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.

Consequence

One consequence of the construction is the below:

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

References

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