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Change of fiber

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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 ) × I → I → β B {\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 g : p − 1 ( b ) × I → E {\displaystyle g:p^{-1}(b)\times I\to E} with g 0 : p − 1 ( b ) ↪ E {\displaystyle g_{0}:p^{-1}(b)\hookrightarrow E} . We have:

g 1 : p − 1 ( b ) → p − 1 ( β ( 1 ) ) {\displaystyle g_{1}:p^{-1}(b)\to p^{-1}(\beta (1))} .

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

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

τ : Pc ⁡ ( B ) → {\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

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

Drawing a picture, there is a homeomorphism I 2 → I 2 {\displaystyle I^{2}\to I^{2}} that restricts to a homeomorphism K → I × { 0 } {\displaystyle K\to I\times \{0\}} . Let f : p − 1 ( b ) × K → E {\displaystyle f:p^{-1}(b)\times K\to E} be such that f ( x , s , 0 ) = g ( x , s ) {\displaystyle f(x,s,0)=g(x,s)} , f ( x , s , 1 ) = g ′ ( x , s ) {\displaystyle f(x,s,1)=g'(x,s)} and f ( x , 0 , t ) = x {\displaystyle 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 {\displaystyle p^{-1}(b)\times I^{2}\to I^{2}{\overset {h}{\to }}B} to w such that w restricts to f {\displaystyle f} . In particular, we have g 1 ∼ g 1 ′ {\displaystyle g_{1}\sim g_{1}'} , establishing the claim.

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

τ ( [ c b ] ) = id , τ ( [ β ] ⋅ [ γ ] ) = τ ( [ β ] ) ∘ τ ( [ γ ] ) {\displaystyle \tau ()=\operatorname {id} ,\,\tau (\cdot )=\tau ()\circ \tau ()}

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

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

Also, we have: for each b in B,

τ : π 1 ( B , b ) → {\displaystyle \tau :\pi _{1}(B,b)\to } { | homotopy equivalence f : p − 1 ( b ) → p − 1 ( b ) {\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.

Consequence

One consequence of the construction is the below:

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

References

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