Euler characteristic and genus
Counting vertices, edges and faces reveals a global invariant of a closed orientable surface.
$$\chi=V-E+F=2-2g$$
What this means
Counting vertices, edges and faces reveals a global invariant of a closed orientable surface.
Deep dive: mathematical and physical mechanism
For a cell decomposition, the alternating count $V-E+F$ is independent of the triangulation. The genus $g$ counts handles; a sphere has $\chi=2$ and a torus has $\chi=0$. In topological order, the spatial genus can affect the ground-state degeneracy.
How it appears in quantum phenomena
Certain fractional quantum Hall phases exhibit topology-dependent degeneracy on a torus.
Assumptions, caveats and limitations
Euler characteristic alone does not classify all topological spaces.
Connect the ideas
For context, see the Foundations of Topology topic guide and the reading references.