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We demonstrate dimensional ladder of higher dimensional quantum Hall effectsby exploiting quantum Hall effects on arbitrary odd dimensional spheres.Non-relativistic and relativistic Landau models are analyzed on $S^{2k-1}$ inthe $SO(2k-1)$ monopole background. The total sub-band degeneracy of the odddimensional lowest Landau level is shown to be equal to the winding number fromthe base-manifold $S^{2k-1}$ to the one-dimension higher $SO(2k)$ gauge group.Based on the chiral Hopf maps, we clarify the underlying quantum Nambu geometryfor odd dimensional quantum Hall effect and the resulting quantum geometry isnaturally embedded also in one-dimension higher quantum geometry. An origin ofsuch dimensional ladder connecting even and odd dimensional quantum Halleffects is illuminated from a viewpoint of the spectral flow ofAtiyah-Patodi-Singer index theorem in differential topology. We also present aBF topological field theory as an effective field theory in which membraneswith different dimensions undergo non-trivial linking in odd dimensional space.Finally, an extended version of the dimensional hierarchy for higherdimensional quantum Hall liquids is proposed, and its relationship to quantumanomaly and D-brane physics is discussed.

Higher (Odd) Dimensional Quantum Hall Effect and Extended Dimensional Hierarchy