  We introduce a novel gauge-invariant, quantized interband index in
two-dimensional (2D) multiband systems. It provides a bulk topological
classification of a submanifold of parameter space (e.g., an electron valley in
a Brillouin zone), and therefore overcomes difficulties in characterizing
topology of submanifolds. We confirm its topological nature by numerically
demonstrating a one-to-one correspondence to the valley Chern number in $k\cdot
p$ models (e.g., gapped Dirac fermion model), and the first Chern number in
lattice models (e.g., Haldane model). Furthermore, we derive a band-resolved
topological charge and demonstrate that it can be used to investigate the
nature of edge states due to band inversion in valley systems like multilayer
graphene.
