  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.
