[1] The concept of topological quantum numbers being created or destroyed during phase transitions emerged in condensed matter physics in the 1970s.The Kosterlitz-Thouless Transition demonstrated how topological defects, like vortices, could be created and annihilated during phase transitions in two-dimensional systems.
[2] Concurrently, in quantum field theory the 't Hooft-Polyakov monopole model demonstrated how topological structures, such as magnetic monopoles, could appear or disappear depending on the phase of a field, linking phase transitions to shifts in topological quantum numbers.
By taking real three-dimensional space, and closing it with a point at infinity, one also gets a 3-sphere.
Solutions to Skyrme's equations in real three-dimensional space map a point in "real" (physical; Euclidean) space to a point on the 3-manifold SU(2).
In the above example, the topological statement is that the 3rd homotopy group of the three sphere is and so the baryon number can only take on integer values.
The one-dimensional sine-Gordon equation makes for a particularly simple example, as the fundamental group at play there is and so is literally a winding number: a circle can be wrapped around a circle an integer number of times.
An example includes screw-type dislocations associated with Germanium whiskers.