Manifolds, spheres and tori
A manifold looks locally like Euclidean space, even when its global topology is different.
$$\mathrm{BZ}_{d}\cong T^d$$
What this means
A manifold looks locally like Euclidean space, even when its global topology is different.
Deep dive: mathematical and physical mechanism
A two-dimensional Brillouin zone identifies opposite edges of a momentum-space cell, producing a torus $T^2$. A normalized two-level Bloch vector lies on $S^2$. Maps from the momentum torus to the Bloch sphere can carry integer degree.
How it appears in quantum phenomena
Two-band Chern insulators map each crystal momentum to a point on the Bloch sphere.
Assumptions, caveats and limitations
A torus in momentum space is not necessarily a doughnut-shaped real-space sample.
Connect the ideas
For context, see the Foundations of Topology topic guide and the reading references.