Change of fiber

Definition

If β is a path in B that starts at, say, b, then we have the homotopy h : p − 1 ( b ) × I → I → β 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 ) × I → E with g 0 : p − 1 ( b ) ↪ E . We have:

g 1 : p − 1 ( b ) → p − 1 ( β ( 1 ) ) .

(There might be an ambiguity and so β ↦ g 1 need not be well-defined.)

Let Pc ⁡ ( B ) denote the set of path classes in B. We claim that the construction determines the map:

τ : Pc ⁡ ( B ) → 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 × { 0 , 1 } ∪ { 0 } × I ⊂ I 2 .

Drawing a picture, there is a homeomorphism I 2 → I 2 that restricts to a homeomorphism K → I × { 0 } . Let f : p − 1 ( b ) × K → 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 ) × I 2 → I 2 → h B to w such that w restricts to f . In particular, we have g 1 ∼ g 1 ′ , establishing the claim.

It is clear from the construction that the map is a homomorphism: if γ ( 1 ) = β ( 0 ) ,

τ ( [ c b ] ) = id , τ ( [ β ] ⋅ [ γ ] ) = τ ( [ β ] ) ∘ τ ( [ γ ] )

where c b is the constant path at b. It follows that τ ( [ β ] ) has inverse. Hence, we can actually say:

τ : Pc ⁡ ( B ) → the set of homotopy classes of homotopy equivalences.

Also, we have: for each b in B,

τ : π 1 ( B , b ) → { [ƒ] | homotopy equivalence f : p − 1 ( b ) → 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.