  We generalize the flux insertion argument due to Laughlin,
Niu-Thouless-Tao-Wu, and Avron-Seiler-Zograf to the case of fractional quantum
Hall states on a higher-genus surface. We propose this setting as a test to
characterise the robustness, or topologicity, of the quantum state of matter
and apply our test to the Laughlin states. Laughlin states form a vector
bundle, the Laughlin bundle, over the Jacobian - the space of Aharonov-Bohm
fluxes through the holes of the surface. The rank of the Laughlin bundle is the
degeneracy of Laughlin states or, in presence of quasiholes, the dimension of
the corresponding full many-body Hilbert space; its slope, which is the first
Chern class divided by the rank, is the Hall conductance. We compute the rank
and all the Chern classes of Laughlin bundles for any genus and any number of
quasiholes, settling, in particular, the Wen-Niu conjecture. Then we show that
Laughlin bundles with non-localized quasiholes are not projectively flat and
that the Hall current is precisely quantized only for the states with localized
quasiholes. Hence our test distinguishes these states from the full many-body
Hilbert space.
