Non-commutativity of exponential spectrum

In a Banach algebra, the spectrum satisfies $\sigma(ab)\setminus{0} = \sigma(ba)\setminus{0}$foreachpairofelements$a,b$. We show that this is no longer true for the exponential spectrum, thereby solving a problem open since 1992. Our proof depends on the fact that the homotopy group ${\pi }_4(G{L}_2(\mathbb{C}))$ is non-trivial.