  We report on the experimental observation of a gapped nontrivial Euler
topology in an acoustic metamaterial. The topological invariant in this case is
the Euler class, which characterizes the two-dimensional geometry of the
acoustic Bloch wave function of multiple bands in the Brillouin zone -- a
feature that cannot be captured by conventional topological classification
schemes. Starting from a carefully designed acoustic system, we retrieve a rich
phase diagram that includes the gapped Euler phase on top of non-Abelian
topological nodal phases. In particular we find that the gapped phase is
characterized by a meron (half-skyrmion) type topology in which the
cancellation of the orbital Zak phases by Zak phases of the bands contributes
in forming the characterizing winding number of the acoustic bands. Using
pump-probe techniques, we are able to extract the wave functions of the
acoustic Bloch bands and demonstrate the non-trivial Euler topology via a
direct observation of the meron geometry of these bands in the Brillouin zone,
in addition to the detection of the acoustic bulk and edge dispersions. We
finally observe a topological edge state in the gapped Euler phase which is a
direct manifestation of the underlying cancellation of the Zak phases. These
experimental features reveal the unprecedented topological Euler phase as well
as setting a benchmark for probing acoustic wave function geometry in the
Brillouin zone.
