  The iron chalcogenides have been proved to be intrinsic topological
superconductors to implement quantum computation because of their unique
electronic structures. The topologically nontrivial surface states of
FeTe$_{0.5}$Se$_{0.5}$ have been predicted by several calculations and then
confirmed by high-resolution photoemission and scanning tunneling experiments.
However, so far, the shreds of the electrical transport evidence for
topological surface states are still in absence. By carrying out electrical
transport experiments, we observe a topological transition with a nonlinear
Hall conductivity and simultaneous linear magnetoresistance near the
superconducting transition temperature. Furthermore, we observe a sign reversal
of the Hall coefficient accompanied by a concurrently softening of the
${A}_{1g}$ phonon mode at about 40 K, indicating a nematic transition. The
synchronized phonon softening with nematicity manifests an enhanced fluctuation
state through spin-phonon interaction. Our results solidly corroborate the
topological surface states of FeTe$_{0.5}$Se$_{0.5}$ and provide an
understanding of the mechanism of the superconductivity in iron chalcogenides.
