Title: Reduction of topological $\mathbb{Z}$ classification in cold atomic systems,
Abstract: One of the most challenging problems in correlated topological systems is a
realization of the reduction of topological classification, but very few
experimental platforms have been proposed so far. We here demonstrate that
ultracold dipolar fermions (e.g., $^{167}$Er, $^{161}$Dy, and $^{53}$Cr) loaded
in an optical lattice of two-leg ladder geometry can be the first promising
testbed for the reduction $\mathbb{Z}\to\mathbb{Z}_4$, where solid evidence for
the reduction is available thanks to their high controllability. We further
give a detailed account of how to experimentally access this phenomenon; around
the edges, the destruction of one-particle gapless excitations can be observed
by the local radio frequency spectroscopy, while that of gapless spin
excitations can be observed by a time-dependent spin expectation value of a
superposed state of the ground state and the first excited state. We clarify
that even when the reduction occurs, a gapless edge mode is recovered around a
dislocation, which can be another piece of evidence for the reduction.