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