Non-commutativity of exponential spectrum

In a Banach algebra, the spectrum satisfies $\sigma(ab)\setminus\{0\} = \sigma(ba)\setminus\{0\}$ for each pair of elements $a,b$. 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 $\pi_4(GL_2({\mathbb C}))$ is non-trivial.