  Motivated by the $\Omega$-spectrum proposal of unique gapped ground states by
Kitaev, we study adiabatic cycles in gapped quantum spin systems from various
perspectives. We give a few exactly solvable models in one and two spatial
dimensions and discuss how nontrivial adiabatic cycles are detected. For one
spatial dimension, we study the adiabatic cycle in detail with the matrix
product state and show that the symmetry charge can act on the space of
matrices without changing the physical states, which leads to nontrivial loops
with symmetry charges. For generic spatial dimensions, based on the Bockstein
isomorphism $H^d(G,U(1)) \cong H^{d+1}(G,\mathbb{Z})$, we study a group
cohomology model of the adiabatic cycle that pumps a symmetry-protected
topological phase on the boundary by one period. It is shown that the spatial
texture of the adiabatic Hamiltonian traps a symmetry-protected topological
phase in one dimension lower.
