  The last years have witnessed rapid progress in the topological
characterization of out-of-equilibrium systems. We report on robust signatures
of a new type of topology -- the Euler class -- in such a dynamical setting.
The enigmatic invariant $(\xi)$ falls outside conventional symmetry-eigenvalue
indicated phases and, in simplest incarnation, is described by triples of bands
that comprise a gapless pair, featuring $2\xi$ stable band nodes, and a gapped
band. These nodes host non-Abelian charges and can be further undone by
converting their charge upon intricate braiding mechanisms, revealing that
Euler class is a fragile topology. We theoretically demonstrate that quenching
with non-trivial Euler Hamiltonian results in stable monopole-antimonopole
pairs, which in turn induce a linking of momentum-time trajectories under the
first Hopf map, making the invariant experimentally observable. Detailing
explicit tomography protocols in a variety of cold-atom setups, our results
provide a basis for exploring new topologies and their interplay with
crystalline symmetries in optical lattices beyond paradigmatic Chern
insulators.
