  Fermi-surface topology governs the relationship between magnetism and
superconductivity in iron-based materials. Using low-temperature transport,
angle-resolved photoemission, and x-ray diffraction we show unambiguous
evidence of large Fermi surface reconstruction in CaFe$_{2}$As$_{2}$ at
magnetic spin-density-wave and nonmagnetic collapsed-tetragonal ($cT$)
transitions. For the $cT$ transition, the change in the Fermi surface topology
has a different character with no contribution from the hole part of the Fermi
surface. In addition, the results suggest that the pressure effect in
CaFe$_{2}$As$_{2}$ is mainly leading to a rigid-band-like change of the valence
electronic structure. We discuss these results and their implications for
magnetism and superconductivity in this material.
