  We study the quench dynamics of a topologically trivial one-dimensional
gapless wire following its sudden coupling to topological bound states. We find
that as the bound states leak into and propagate through the wire, signatures
of their topological nature survive and remain measurable over a long lifetime.
Thus, the quench dynamically induces topological properties in the gapless
wire. Specifically, we study a gapless wire coupled to fractionally charged
solitons or Majorana fermions and characterize the dynamically induced topology
in the wire, in the presence of disorder and short-range interactions, by
analytical and numerical calculations of the dynamics of fractional charge,
fermion parity, entanglement entropy, and fractional exchange statistics. In a
dual effective description, this phenomenon is described by correlators of
boundary changing operators, which, remarkably, generate topologically
non-trivial monodromies in the gapless wire, both for abelian and non-abelian
quantum statistics of the bound states.
