Theorem (Smale). If N is a C^{2} manifold of dimension at least three, and F_{2}(N) is the bundle of two-frames of N, the regular homotopy classes of C^{2} immersions of S^{2} in N based at y_{0} ∈ F_{2}(N) are in one-to-one correspondence with elements of π_{2}(F_{2}(N), y_{0}).

Take the case N = ℝ^{3}. Since F_{2}(N) is homeomorphic to SO(3), and since π_{2}(SO(3)) = 0, this implies that all immersions of S^{2} in ℝ^{3} are regularly homotopic. In particular, the sphere can be turned inside out by means of regular homotopies.