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 [math]\displaystyle{ h: p^{-1}(b) \times I \to I \overset{\beta}\to B }[/math] where the first map is a projection. Since p is a fibration, by the homotopy lifting property, h lifts to a homotopy [math]\displaystyle{ g: p^{-1}(b) \times I \to E }[/math] with [math]\displaystyle{ g_0: p^{-1}(b) \hookrightarrow E }[/math]. We have:

[math]\displaystyle{ g_1: p^{-1}(b) \to p^{-1}(\beta(1)) }[/math].

(There might be an ambiguity and so [math]\displaystyle{ \beta \mapsto g_1 }[/math] need not be well-defined.)

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

[math]\displaystyle{ \tau: \operatorname{Pc}(B) \to }[/math] 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

[math]\displaystyle{ K = I \times \{0, 1\} \cup \{0\} \times I \subset I^2 }[/math].

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

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

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

[math]\displaystyle{ \tau([c_b]) = \operatorname{id}, \, \tau([\beta] \cdot [\gamma]) = \tau([\beta]) \circ \tau([\gamma]) }[/math]

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

[math]\displaystyle{ \tau: \operatorname{Pc}(B) \to }[/math] the set of homotopy classes of homotopy equivalences.

Also, we have: for each b in B,

[math]\displaystyle{ \tau: \pi_1(B, b) \to }[/math] { [ƒ] | homotopy equivalence [math]\displaystyle{ f : p^{-1}(b) \to p^{-1}(b) }[/math] }

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