  In quantum multiparameter estimation, multiple to-be-estimated parameters are
encoded in a quantum dynamics system by a unitary evolution. As the parameters
vary, the system may undergo a topological phase transition (TPT). In this
paper, we investigate two SU(2) TPT models and propose the singular behavior of
the quantum metric tensor around the TPT point as a tool for the simultaneous
optimal estimation of multiple parameters. We find that the proposed TPT
sensing protocol can achieve the same metrology performance as the
quantum-control-enhanced one. Moreover, the probe state of the TPT sensing
protocol is only the ground state of the Hamiltonian rather than the entangled
state required in the control-enhanced one. In addition, an adaptive
multiparameter estimation strategy is developed for updating the estimated
values until the desired quantum Cram\'er-Rao bound is approached. Our work
reinforces the connection between quantum multiparameter estimation and
topology physics, with potential inspiration for quantum critical metrology.
