  Symmetries play a crucial role in the classification of topological phases of
matter. Although recent studies have established a powerful framework to search
for and classify topological phases based on symmetry indicators, there exists
a large class of fragile topology beyond the description. The Euler class
characterizing the topology of two-dimensional real wave functions is an
archetypal fragile topology underlying some important properties, such as
non-Abelian braiding of crossing nodes and higher-order topology. However, as a
minimum model of fragile topology, the two-dimensional topological Euler
insulator consisting of three bands remains a significant challenge to be
implemented in experiments. Here, we experimentally realize a three-band
Hamiltonian to simulate a topological Euler insulator with a trapped-ion
quantum simulator. Through quantum state tomography, we successfully evaluate
the Euler class, Wilson loop flow and entanglement spectra to show the
topological properties of the Hamiltonian. We also measure the Berry phases of
the lowest energy band, illustrating the existence of four crossing points
protected by the Euler class. The flexibility of the trapped-ion quantum
simulator further allows us to probe dynamical topological features including
skyrmion-antiskyrmion pairs and Hopf links in momentum-time space from quench
dynamics. Our results show the advantage of quantum simulation technologies for
studying exotic topological phases and open a new avenue for investigating
fragile topological phases in experiments.
