In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle π : E → B {\displaystyle \pi :E\rightarrow B} and a continuous map f : B ′ → B {\displaystyle f:B'\rightarrow B} one can define a "pullback" of E {\displaystyle E} by f {\displaystyle f} as a bundle f ∗ E {\displaystyle f^{*}E} over B ′ {\displaystyle B'} . The fiber of f ∗ E {\displaystyle f^{*}E} over a point b ′ {\displaystyle b'} in B ′ {\displaystyle B'} is just the fiber of E {\displaystyle E} over f ( b ′ ) {\displaystyle f(b')} . Thus f ∗ E {\displaystyle f^{*}E} is the disjoint union of all these fibers equipped with a suitable topology.
Formal definition Let π : E → B {\displaystyle \pi :E\rightarrow B} be a fiber bundle with abstract fiber F {\displaystyle F} and let f : B ′ → B {\displaystyle f:B'\rightarrow B} be a continuous map. Define the pullback bundle by
f ∗ E = { ( b ′ , e ) ∈ B ′ × E ∣ f ( b ′ ) = π ( e ) } ⊆ B ′ × E {\displaystyle f^{*}E=\{(b',e)\in B'\times E\mid f(b')=\pi (e)\}\subseteq B'\times E}
and equip it with the subspace topology and the projection map π ′ : f ∗ E → B ′ {\displaystyle \pi ':f^{*}E\rightarrow B'} given by the projection onto the first factor, i.e.,
π ′ ( b ′ , e ) = b ′ . {\displaystyle \pi '(b',e)=b'.\,}
The projection onto the second factor gives a map
h : f ∗ E → E {\displaystyle h\colon f^{*}E\to E}
such that the following diagram commutes:
f ∗ E ⟶ h E π ′ ↓ ↓ π B ′ ⟶ f B {\displaystyle {\begin{array}{ccc}f^{\ast }E&{\stackrel {h}{\longrightarrow }}&E\\{\pi }'\downarrow &&\downarrow \pi \\B'&{\stackrel {f}{\longrightarrow }}&B\end{array}}}
If ( U , φ ) {\displaystyle (U,\varphi )} is a local trivialization of E {\displaystyle E} then ( f − 1 U , ψ ) {\displaystyle (f^{-1}U,\psi )} is a local trivialization of f ∗ E {\displaystyle f^{*}E} where
ψ ( b ′ , e ) = ( b ′ , proj 2 ( φ ( e ) ) ) . {\displaystyle \psi (b',e)=(b',{\mbox{proj}}_{2}(\varphi (e))).\,}
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