Every principal bundle over $\mathbb{R}^n$ is trivial

Question

On page 222 in Naber's "Topology, Geometry and Gauge fields: Foundations" there is the following remark.

Using more general versions of the Homotopy Lifting Theorem one can prove that any principal G-bundle over a contractible, paracompact (see [Dug]) space is trivial. In particular, any principal bundle over $\mathbb{R}^n$ is trivial.

Is there an easier proof of this statement in the case of a principal bundle over $\mathbb{R}^n$? In particular, is it possible to construct a global trivialization or a global section directly?

Answer 1

Let $G$ be a reasonably nice topological group (e.g. a Lie group). Then there exist a topological space $B G$ and a principal $G$-bundle $E G \to B G$ with the following universal property:

• For all paracompact spaces $X$ with the homotopy type of a CW-complex, the map sending a continuous map $f : X \to B G$ to the principal $G$-bundle $f^* E G \to X$ induces a bijection between homotopy classes of continous maps $X \to B G$ and isomorphism classes of principal $G$-bundles on $X$.

In particular, if $X$ is contractible, then there is only one homotopy class of continuous maps $X \to B G$, so every principal $G$-bundle on $X$ is trivial.

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The above argument is potentially circular as one usually starts by proving the following:

Let $X$ be a paracompact space and let $E' \to X \times [0, 1]$ be a principal $G$-bundle. Then the restriction over $X \times \{ 0 \}$ is isomorphic to the restriction over $X \times \{ 1 \}$.

Once this is known, we can immediately deduce that principal $G$-bundles over a contractible paracompact space must be trivial: choose a map $h : X \times [0, 1] \to X$ such that $h (x, 0) = x$ for all $x$, and $h (x, 1) = h (x', 1)$ for all $x$ and $x'$; then for any principal $G$-bundle $E \to X$, we get a principal $G$-bundle $h^* E \to X \times [0, 1]$ whose restriction over $X \times \{ 0 \}$ is $E \times \{ 0 \} \to X \times \{ 0 \}$, while the restriction over $X \times \{ 1 \}$ is trivial.