nLab local trivialization

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:

1. 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.

2. 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

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

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

3. 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:

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

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.