  Fermi gases and liquids display an excitation spectrum that is simply
connected, ensuring closed Fermi surfaces. In strongly correlated systems like
the cuprate superconductors, the existence of open sheets of Fermi surface
known as Fermi arcs indicates a distinctly different topology of the spectrum
with no equivalent in Fermi liquid theory. Here, we demonstrate a generic
mechanism by which correlation effects in fermionic systems can change the
topology of the spectrum. Using diagrammatic Monte Carlo simulations, we
demonstrate the existence of disconnected and multiply connected excitation
spectra in the attractive Hubbard model in the BCS-BEC cross-over regime. These
topologically nontrivial spectra are a prerequisite for Fermi arcs.
