Immersions of Manifolds

Immersions of an m-manifold in an n-manifold, n>m, are classified up to regular homotopy by the homotopy classes of sections of a vector bundle E associated to the tangent bundle of M.  When N = Rn , the fibre of E is the Stiefel manifold of m-frames in n-space.

Immersions of almost parallizable

An almost parallelizable n-manifold M can be immersed in Euclidean q-space if 2q>3n.

Immersions in the stable range

Immersions of a differentiable m-manifold M in a differentiable n-manifold N, 2n > 3m+1, are classified up to regular homotopy by the homotopy classes of fibre maps F: T(M) ----> T(N) such that F(-X)=-F(X) and F(X) is nonzero of X is nonzero.