  In high-temperature cuprate superconductors, the anti-ferromagnetic spin
fluctuations are thought to have a very important role in naturally producing
an attractive interaction between the electrons in the $d$-wave channel. The
connection between superconductivity and spin fluctuations is expected to be
especially consequential at the overdoped end point of the superconducting
dome. In some materials, that point seems to coincide with a Lifshitz
transition, where the Fermi surface changes from the hole-like centered at
($\pi, \pi$) to the electron-like, centered at the $\Gamma$ point causing a
loss of large momentum anti-ferromagnetic fluctuations. Here, we study the
doping dependence of the electronic structure of
Bi$_{1.8}$Pb$_{0.4}$Sr$_2$CuO$_{6+\delta}$ in angle-resolved photoemission and
find that the superconductivity vanishes at lower doping than at which the
Lifshitz transition occurs. This requires a more detailed re-examination of a
spin-fluctuation scenario.
