When are the transition functions of a smooth complex vector bundle homotopic to a constant map?

Question

Let $E \to M$ be a smooth complex vector bundle and $\tau : U \to \mathrm{GL}_n(\mathbb C)$ any of its transition functions. If $U$ is contractible, then of course $\tau$ is homotopic to the constant map $p \in U \mapsto \mathrm{id} \in \mathrm{GL}_n(\mathbb C)$. But what if $U $ not contractible? Could I perhaps use the cocycle condition on the transition functions to show that any of them is homotopic to a constant map?

Answer 1

If $U$ is not simply connected, then the transition function $\tau:U\to\text{GL}(n,\mathbb{C})$ will typically not be homotopic to the constant identity map. So no, you cannot guarantee this. Counterexample: take the $2$-sphere with its usual open cover. The intersection $U$ deformation retracts onto the equator. The corresponding transition function is homotopic to the constant identity map if and only if the bundle over $S^2$ itself is trivial. This is clearly not always the case; there are non-trivial vector bundles over $S^2$.

More generally, if $\tau$ is homotopic to the constant identity map, then $E$ must be trivial over $U_\alpha\cup U_\beta$, where $U=U_\alpha\cap U_\beta$, and the converse is likewise true. See for example Hatcher's book on topological K-theory.