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 $$h: p^{-1}(b) \times I \to I \overset{\beta}\to B$$ where the first map is a projection. Since p is a fibration, by the homotopy lifting property, h lifts to a homotopy $$g: p^{-1}(b) \times I \to E$$ 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 $$\operatorname{Pc}(B)$$ denote the set of path classes in B. We claim that the construction determines the map:
 * $$\tau: \operatorname{Pc}(B) \to $$ 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
 * $$K = I \times \{0, 1\} \cup \{0\} \times I \subset I^2$$.

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 $$f(x, s, 0) = g(x, s)$$, $$f(x, s, 1) = g'(x, s)$$ and $$f(x, 0, t) = x$$.

Then, by the homotopy lifting property, we can lift the homotopy $$p^{-1}(b) \times I^2 \to I^2 \overset{h}\to B$$ to w such that w restricts to $$f$$. In particular, we have $$g_1 \sim g_1'$$, establishing the claim.

It is clear from the construction that the map is a homomorphism: if $$\gamma(1) =\beta(0)$$,
 * $$\tau([c_b]) = \operatorname{id}, \, \tau([\beta] \cdot [\gamma]) = \tau([\beta]) \circ \tau([\gamma])$$

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