  Traditional photonuclear reactions primarily excite giant dipole resonances,
making the measurement of isovector giant resonances with higher multipolarties
a great challenge. In this work, the manipulation of collective excitations of
different multipole transitions in nuclei via vortex $\gamma$ photons has been
investigated. We develop the calculation method for photonuclear cross sections
induced by the vortex $\gamma$ photon beam using the fully self-consistent
random-phase approximation plus particle-vibration coupling (RPA+PVC) model
based on Skyrme density functional. We find that the electromagnetic
transitions with multipolarity $J< m_\gamma$ are forbidden for vortex $\gamma$
photons due to the angular momentum conservation, with $m_\gamma$ being the
projection of total angular momentum of $\gamma$ photon on its propagation
direction. For instance, this allows for probing the isovector giant quadrupole
resonance without interference from dipole transitions using vortex $\gamma$
photons with $m_\gamma=2$. The electromagnetic transitions with $J>m_\gamma$
are strongly suppressed compared with the plane-wave-$\gamma$-photon case, and
even vanish at specific polar angles. Therefore, the giant resonances with
specific multipolarity can be extracted via vortex $\gamma$ photons. Moreover,
the vortex properties of $\gamma$ photons can be meticulously diagnosed by
measuring the nuclear photon-absorption cross section. Our method opens new
avenues for photonuclear excitations, generation of coherent $\gamma$ photon
laser and precise detection of vortex particles, and consequently, has
significant impact on nuclear physics, nuclear astrophysics and strong laser
physics.
