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.

On imbedding differentiable manifolds in Euclidean space

Assume n,k,m,q are positive integers. Let M^n denote a smooth differentiable n-manifold and R^k Euclidean k-space. (a) If M^n is open it imbeds smoothly in R^k, k=2n-1 (b) If M^n is open and parallelizable it immerses in R^n (c) Assume M^n is closed and (m-1)-connected, 1< 2m-n < n+1. If a neighborhood of the (n-m)-skeleton immerses in R^q, a>2n-2m, then the complement of a point of M^n imbeds smoothly in R^q.

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.

Smooth regular neighborhoods

If K is a subcomplex of a smooth triangulation of a manifold M and N is a regular neighborhood of K contained in an open neighborhood U of K in M, there is a piecewise regular homeomorphism f of M carrying N onto a smooth submanifold of U, and f reduces to the identity map on K and outside U.

On homotopy spheres of low dimension

Let M be a smooth compact manifold homotopy equivalent to the 4-sphere S^4. Then M x R^1 is homeomorphism co S^4 x R^1.