nLab
local trivialization
Contents
Contents
Idea
In the theory of fiber bundles (including principal bundles , vector bundles , etc.) internal to TopologicalSpaces or SmoothManifolds etc., the condition that these be locally trivial is typically crucial. This means that locally, i.e. over a neighbourhood of any point in their base space, these are equivalent to a trivial fiber bundle , hence to the projection out of the Cartesian product of that neighbourhood with a fiber -space (the typical fiber ).

Key implications of local triviality are:

the induced bundle isomorphisms between local trivializations on intersections of their open neighbourhoods give a system of transition functions which constitute the representation of the given fiber bundle as a cocycle in non-abelian Cech cohomology .

the fact that Serre fibrations are detected locally (see there ) implies that any local fiber bundle is a Serre fibration (and even a Hurewicz fibration if it is a numerable fiber bundle ).

Definition
For topological fiber bundles
For $p \colon E \to X$ a bundles in TopologicalSpaces , hence a continuous function between topological spaces , then a local trivialization is

an open cover $\mathcal{U} \coloneqq \underset{i \in I}{\sqcup} U_i \longrightarrow X$

a topological space $F$ , to be called the typical fiber ;

an isomorphism over $\mathcal{U}$ between the restriction of $E$ to $\mathcal{U}$ and the projection out of the Cartesian product of the cover with the typical fiber:

$\array{
U \times F
&\underoverset{t}{\simeq}{\longrightarrow}&
E|_{U}&\longrightarrow& E
\\
&
\searrow
&
\downarrow &{}^{{}_{(pb)}}& \downarrow^{\mathrlap{p}}
\\
&&
\mathcal{U} &\longrightarrow& X
}$

If this exist, then in particular the total square here is a pullback square , and one says that $p$ is locally trivializable .

For equivariant topological fiber bundles
For equivariant bundles one typically needs a slightly more sophisticated notion, see there .

References
See the references at fiber bundle .

Last revised on June 3, 2022 at 00:29:25.
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