  Floquet engineering is a powerful manipulation method in modern quantum
technology. However, unwanted heating is the main challenge of Floquet
engineering, therefore the Floquet thermalization has attracting considerable
attentions recently. In this work, we investigate thermalization of
periodically driven many-body systems through the lens of Krylov complexity,
and find a topological origin of different thermalization behaviors. We
demonstrate that If the topology of the Krylov chain is nontrivial, a
periodically driven system will reach a state with finite temperature. When the
Krylov chain is topologically trivial, the system will be heated to infinite
temperature. We further show that the prethermalization can be understood as
the tunnelling process of a quasi-edge mode through the local gap on Krylov
chain. This picture provides a systematically method to obtain the effective
prethermal Hamiltonian.
