Applications of Principal Bundle Construction: Vague Question

Question

I recently read the principal $G$-bundle construction on a smooth manifold $M$, where $G$ is a Lie group.

To understand them better, I am looking for some applications.

Can the principal $G$-bundle help us get some usual bundle constructions, for example tensor product of two vector bundles, the pullback bundle etc?

Right now, the constructions I have seen are specific to each type of construction. If I want the tensor product of two vector bundles $E$ and $F$ over a smooth manifold $M$, I start from scratch and consider the disjoint union $\bigsqcup_{p\in M} E_p\otimes F_p$ and put trivializations "naturally". Similarly for the dual bundle.

Is there a unified way to think of these constructions so that all the constructions are dealt with in one shot?

Answer 1

If $f : G \to H$ is a morphism of Lie groups, it induces a functor from principal $G$-bundles to principal $H$-bundles (explicitly, apply $f$ to Cech cocycles). This construction is itself functorial. This subsumes the usual bundle constructions. For example:

• Direct sum comes from morphisms $GL_n \times GL_m \to GL_{n+m}$.
• Tensor product comes from morphisms $GL_n \times GL_m \to GL_{nm}$.
• Dual comes from the inverse transpose morphisms $GL_n \to GL_n$.
• "Underlying real bundle" comes from morphisms $GL_n(\mathbb{C}) \to GL_{2n}(\mathbb{R})$.
• "Complexification" comes from morphisms $GL_n(\mathbb{R}) \to GL_n(\mathbb{C})$.

And so forth.

But the idea of principal bundles has many other applications. Some have a more homotopy-theoretic flavor, e.g. the theory of characteristic classes and reduction of the structure group, while others have a more differential-geometric flavor, e.g. connections and Chern-Weil theory.