Although fragile topology has been intensely studied in static crystals, it is not clear how to generalize the concept to dynamical systems. In this work, we generalize the concept of fragile topology, and provide a definition of fragile topology for noninteracting Floquet crystals, which we refer to as dynamical fragile topology. In contrast to the static fragile topology defined for Wannier obstruction, dynamical fragile topology is defined for the nontrivial quantum dynamics characterized by obstruction to static limits (OTSL). Specifically, OTSL of a Floquet crystal is fragile if and only if the OTSL disappears after adding a symmetry-preserving static Hamiltonian in a direct-sum way preserving the relevant gaps (RGs). We further present a concrete 2+1D example for dynamical fragile topology, based on a slight modification of the model in [Rudner et al, Phys. Rev. X 3, 031005 (2013)]. The fragile OTSL in the 2+1D example exhibits anomalous chiral edge modes for a natural open boundary condition, and does not require any crystalline symmetries besides lattice translations. Our work paves the way to study fragile topology for general quantum dynamics.
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