Homotopy and winding
A loop can wrap a circle an integer number of times; smooth deformations cannot change that integer without breaking continuity.
$$\pi_1(S^1)\cong\mathbb Z$$
What this means
A loop can wrap a circle an integer number of times; smooth deformations cannot change that integer without breaking continuity.
Deep dive: mathematical and physical mechanism
For a map $q:S^1\to S^1$, the winding number is $\nu=(2\pi i)^{-1}\oint q^{-1}dq$. Its integer character follows from the net phase change around a closed loop. Higher homotopy groups describe maps from higher-dimensional spheres.
How it appears in quantum phenomena
The Su–Schrieffer–Heeger chain realizes a one-dimensional winding number.
Assumptions, caveats and limitations
Winding can require chiral symmetry and a nonzero gap.
Connect the ideas
For context, see the Foundations of Topology topic guide and the reading references.