On the topological charge conservation in the three-dimensional ${\rm O}(3)$ $\sigma$-model.

summary:A field of three-component unit vectors on the $2+1$ dimensional spacetime is considered. Two field configurations with different values of the topological charge cannot be connected by the path of field configurations with a finite Euclidean action. Therefore there is no transition between them. The initial and final configurations are assumed to be continuous at infinity. The asymptotic behaviour of intermediate configurations may be arbitrary. The proof is based on the properties of the degree of mapping