Dimensional extension by adiabatic phase evolution connects one-dimensional (1D) Peierls chains to two-dimensional (2D) Chern insulators and identifies chiral solitons as the 1D analogs of quantum Hall edge states. Here, we show that this framework cannot generate the 1D counterparts of quantum spin-Hall edge states, which we call helical solitons, within the class of time-reversal (TR)-invariant extensions built from real-valued order parameters. For $s_z$-conserving models, the spin-Chern numbers of the dimensionally extended 2D system satisfy $C_{\uparrow }=C_{\downarrow }$, ruling out the $C_{\uparrow }=-C_{\downarrow }$ structure required for quantum spin-Hall topology. For spin-mixing Rashba coupling, the same obstruction holds for the projected spin-Chern numbers provided the spin-spectrum gap remains open. The origin is that the adiabatic parameter is not a physical momentum and therefore does not flip under time reversal, removing one sign factor from the Berry curvature transformation. The result constrains topological pumping in TR-invariant Peierls systems and has direct implications for indium atomic wires and related surface systems.

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• Edge states
• Spin-orbit coupling
• Symmetry protected topological states
• Topological edge modes
• Topological materials