The question of whether two gauge configurations can be smoothly deformed into each other is formally described by the homotopy group of the mapping \Omega(x): S^3 \rightarrow G. For example, the gauge group G=\text{SU}(2) has an underlying manifold of S^3 so that the mapping is \Omega(x):S^3 \rightarrow S^3, which has a homotopy group of \pi_3(\text{SU}(2)) = \mathbb Z. This means that every mapping has some integer associated with it called its winding number, also known as its Pontryagin index, with it roughly describing to how many times the spatial S^3 is mapped onto the group S^3, with negative windings occurring due to a flipped orientation. Only mappings with the same winding number can be smoothly deformed into each other and are said to belong to the same homotopy class. Gauge transformations which preserve the winding number are called small gauge transformations while ones that change the winding number are called large gauge transformations.
