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In this Article we address the definition and values of topological numbersof the manifolds of wavefunctions - bands obtained by quantum superposition ofthe wavefunctions that belong to topologically distinct manifolds. The problem,although simple in essence, can be formulated as a paradox: it may seem thatquantum superposition implies non-integer topological numbers.  We show that the results are different for superpositions that are createddynamically and for those obtained by stationary mixing of the manifolds. Fordynamical superpositions we have found that the observable related to thetopological number is non-integer indeed. For static superpositions theresulting bands bear integer topological numbers. We illustrate how the numberquantization is restored upon avoided crossing of topologically distinctsubbands. The crossings lead to the exchange of topological numbers between thebands. We consider complicated phase diagrams arising from this and show theabsence of triple critical points and abundance of quadruple critical pointsthat are rare in thermodynamic phase diagrams. We illustrate these featuresconsidering the bilayer Haldane model.

Topological numbers of quantum superpositions of topologically non-trivial bands